problems in probability theory, mathematical statistics and theory of random functions

ProblemsinProbabilityTheoryMathematicalStatisticsandTheoryofRandomFunctions


EditedbyAASVESHNIKOV

TranslatedbyScriptaTechnicaIncEditedbyBernardRGelbaum

DOVERPUBLICATIONSINCNEWYORK

Copyrightcopy1968byDrRichardASilvermanAllrightsreservedunderPanAmericanandInternationalCopyright

Conventions

ThisDovereditionfirstpublishedin1978isanunabridgedandunalteredrepublicationoftheEnglishtranslationoriginallypublishedbyWBSaundersCompanyin1968

TheworkwasoriginallypublishedbytheNaukaPressMoscowin1965underthetitleSbornikzadachpoteoriiveroyatnosteymatematicheskoystatistikeiteoriisluchaynykhfunktsiy

InternationalStandardBookNumber0-486-63717-4LibraryofCongressCatalogCardNumber78-57171

ManufacturedintheUnitedStatesofAmericaDoverPublicationsInc31East2ndStreetMineolaNY11501

ForewordStudentsatalllevelsofstudyinthetheoryofprobabilityandinthetheoryofstatisticswillfindinthisbookabroadanddeepcross-sectionofproblems(andtheirsolutions)rangingfromthesimplestcombinatorialprobabilityproblemsinfinitesamplespacesthroughinformationtheorylimittheoremsandtheuseofmoments

Theintroductionstothesectionsineachchapterestablishthebasicformulasand notation and give a general sketch of that part of the theory that is to becoveredbytheproblemstofollowPrecedingeachgroupofproblemstherearetypicalexamplesandtheirsolutionscarriedoutingreatdetailEachoftheseiskeyed to the problems themselves so that a student seeking guidance in thesolution of a problem can by checking through the examples discover theappropriatetechniquerequiredforthesolution

BernardRGelbaum

ContentsI RANDOMEVENTS

1 Relationsamongrandomevents

2 Adirectmethodforevaluatingprobabilities

3 Geometricprobabilities

4 ConditionalprobabilityThemultiplicationtheoremforprobabilities

5 Theadditiontheoremforprobabilities

6 Thetotalprobabilityformula

7 Computationoftheprobabilitiesofhypothesesafteratrial(Bayesrsquoformula)

8 Evaluationofprobabilitiesofoccurrenceofaneventinrepeatedindependenttrials

9 ThemultinomialdistributionRecursionformulasGeneratingfunctions

II RANDOMVARIABLES

10 Theprobabilitydistributionseriesthedistributionpolygonandthedistributionfunctionofadiscreterandomvariable

11 Thedistributionfunctionandtheprobabilitydensityfunctionofacontinuousrandomvariable

12 Numericalcharacteristicsofdiscreterandomvariables

13 Numericalcharacteristicsofcontinuousrandomvariables

14 Poissonrsquoslaw

15 Thenormaldistributionlaw

16 Characteristicfunctions

17 Thecomputationofthetotalprobabilityandtheprobabilitydensity

intermsofconditionalprobability

III SYSTEMSOFRANDOMVARIABLES

18 Distributionlawsandnumericalcharacteristicsofsystemsofrandomvariables

19 ThenormaldistributionlawintheplaneandinspaceThemultidimensionalnormaldistribution

20 Distributionlawsofsubsystemsofcontinuousrandomvariablesandconditionaldistributionlaws

IV NUMERICALCHARACTERISTICSANDDISTRIBUTIONLAWSOFFUNCTIONSOFRANDOMVARIABLES

21 Numericalcharacteristicsoffunctionsofrandomvariables

22 Thedistributionlawsoffunctionsofrandomvariables

23 Thecharacteristicfunctionsofsystemsandfunctionsofrandomvariables

24 Convolutionofdistributionlaws

25 Thelinearizationoffunctionsofrandomvariables

26 Theconvolutionoftwo-dimensionalandthree-dimensionalnormaldistributionlawsbyuseofthenotionofdeviationvectors

V ENTROPYANDINFORMATION

27 Theentropyofrandomeventsandvariables

28 Thequantityofinformation

VI THELIMITTHEOREMS

29 Thelawoflargenumbers

30 ThedeMoivre-LaplaceandLyapunovtheorems

VII THECORRELATIONTHEORYOFRANDOMFUNCTIONS

31 Generalpropertiesofcorrelationfunctionsanddistributionlawsofrandomfunctions

32 Linearoperationswithrandomfunctions

33 Problemsonpassages

34 Spectraldecompositionofstationaryrandomfunctions

35 Computationofprobabilitycharacteristicsofrandomfunctionsattheoutputofdynamicalsystems

36 Optimaldynamicalsystems

37 Themethodofenvelopes

VIII MARKOVPROCESSES

38 Markovchains

39 TheMarkovprocesseswithadiscretenumberofstates

40 ContinuousMarkovprocesses

IX METHODSOFDATAPROCESSING

41 Determinationofthemomentsofrandomvariablesfromexperimentaldata

42 Confidencelevelsandconfidenceintervals

43 Testsofgoodness-of-fit

44 Dataprocessingbythemethodofleastsquares

45 Statisticalmethodsofqualitycontrol

46 Determinationofprobabilitycharacteristicsofrandomfunctionsfromexperimentaldata

ANSWERSANDSOLUTIONS

SOURCESOFTABLESREFERREDTOINTHETEXT

BIBLIOGRAPHY

INDEX

I RANDOMEVENTS

1 RELATIONSAMONGRANDOMEVENTS

BasicFormulasRandomeventsareusuallydesignatedbythelettersABChellipUVwhere

UdenotesaneventcertaintooccurandVanimpossibleeventTheequalityA=B means that the occurrence of one of the events inevitably brings about theoccurrenceoftheotherTheintersectionoftwoeventsAandBisdefinedastheeventC=ABsaidtooccurifandonlyifbotheventsAandBoccurTheunionoftwoeventsAandBistheeventC=AcupBsaidtooccurifandonlyifatleastone of the events A andB occurs The difference of two events A and B isdefinedastheevent saidtooccurifandonlyifAoccursandBdoesnotoccurThecomplementary event is denotedby the same letter as theinitialeventbutwithanoverbarForinstance andAarecomplementarymeaningthatAdoesnotoccurTwoeventsaresaidtobemutuallyexclusiveifAB=VTheeventsAk (k=1 2hellipn) are said to formacomplete set if the

experimentresultsinatleastoneoftheseeventssothat

SOLUTIONFORTYPICALEXAMPLES

Example11WhatkindofeventsAandBwillsatisfytheequalityAcupB=A

SOLUTION The unionAcupB means the occurrence of at least one of theeventsAandBThenforAcupB=AtheeventAmustincludetheeventBForexampleifAmeansfallingintoregionSAandBfallingintoregionSBthenSBlieswithinSA

ThesolutiontoProblems11to13and18issimilar

Example12Twonumbersat randomare selected froma tableof random

numbersIftheeventAmeansthatatleastoneofthesenumbersisprimeandtheeventB that at least one of them is an even number what is themeaning ofeventsABandAcupB

SOLUTIONEventABmeansthatbotheventsAandBoccurTheeventAcupBmeans that at least one of the two events occurs that is from two selectednumbersatleastonenumberisprimeoroneisevenoronenumberisprimeandtheotheriseven

OnecansolveProblems14to17analogously

Example13Provethat and

PROOFIfC= andD= thesecondequalitycanbewrittenintheformHenceitsufficestoprovethevalidityofthefirstequality

The event means that both events A and B do not occur Thecomplementary event means that at least one of these events occurs theunionAcupB Thus The proof of this equality can also becarried out geometrically an event meaning that a point falls into a certainregion

OnecansolveProblem19similarlyTheequalitiesprovedinExample13areusedinsolvingProblems110to114

Example14The schemeof anelectric circuitbetweenpointsMandN isrepresentedinFigure1LettheeventAbethattheelementaisoutoforderandlet theeventsBk (k = 1 2 3) be that an elementbk is out of orderWrite theexpressionsforCand wheretheeventCmeansthecircuitisbrokenbetweenMandN

SOLUTION The circuit is broken betweenM andN if the elementa or thethreeelementsbk(k=123)areoutoforderThecorrespondingeventsareAandB1B2B3HenceC=AcupB1B2B3

UsingtheequalitiesofExample13wefindthat

SimilarlyonecansolveProblems116to118

PROBLEMS

11WhatmeaningcanbeassignedtotheeventsAcupAandAA12WhendoestheequalityAB=Ahold13Atargetconsistsof10concentriccirclesofradiusrk(k=123hellip10)

AneventAkmeanshittingtheinteriorofacircleofradiusrk(k=12hellip10)Whatdothefollowingeventsmean

14 Consider the following events A that at least one of three devicescheckedisdefectiveandBthatalldevicesaregoodWhatisthemeaningoftheevents(a)AcupB(b)AB

FIGURE1

15 The events A B andC mean selecting at least one book from threedifferentcollectionsofcompleteworkseachcollectionconsistsofatleastthreevolumes The eventsAs andBk mean that s volumes are taken from the firstcollection andk volumes from the second collection Find themeaning of theevents(a)AcupBcupC(b)ABC(c)A1cupB3(d)A2B2(e)(A1B3cupB1A3)C

16AnumberisselectedatrandomfromatableofrandomnumbersLettheeventAbethatthechosennumberisdivisibleby5andlettheeventBbethatthechosennumberendswithazeroFindthemeaningoftheevents and

17LettheeventAbethatatleastoneoutoffouritemsisdefectiveandlet

theeventBbe thatat least twoof themaredefectiveFindthecomplementaryevents and

18Simplifytheexpression 19Whendothefollowingequalitiesholdtrue(a)AcupB= (b)AB=

(c)AcupB=AB110FromthefollowingequalityfindtherandomeventX

111Provethat112Provethatthefollowingtwoequalitiesareequivalent

113CantheeventsAand besimultaneous114ProvethatA Band formacompletesetofevents115TwochessplayersplayonegameLettheeventAbethatthefirstplayer

winsandletBbe that thesecondplayerwinsWhateventshouldbeaddedtotheseeventstoobtainacompleteset

116AninstallationconsistsoftwoboilersandoneengineLettheeventAbethattheengineisingoodconditionletBk(k=12)bethatthekthboilerisingoodconditionandletCbethattheinstallationcanoperateiftheengineandatleast one of the boilers are in good conditionExpress the eventsC and intermsofAandBk

117AvesselhasasteeringgearfourboilersandtwoturbinesLettheeventAbethatthesteeringgearisingoodconditionletBk(k=1234)bethattheboilerlabeledkisingoodconditionletCj(j=12)bethattheturbinelabeledjisingoodconditionandletDbethatthevesselcansailiftheengineatleastoneoftheboilersandatleastoneoftheturbinesareingoodconditionExpressDand intermsofAandBk

118Adevice ismadeof twounits of the first type and threeunits of thesecond typeLetAk (k = 1 2) be that the kth unit of the first type is in goodcondition letBj (j=123)be that the jthunitof the second type is ingoodconditionandletCbethatthedevicecanoperateifatleastoneunitofthefirsttypeandatleasttwounitsofthesecondtypeareingoodconditionExpresstheeventCintermsofAkandBj

2 ADIRECTMETHODFOREVALUATING

PROBABILITIES

BasicFormulas

Iftheoutcomesofanexperimentformafinitesetofnelementsweshallsaythattheoutcomesareequallyprobableiftheprobabilityofeachoutcomeis1nThusifaneventconsistsofmoutcomestheprobabilityoftheeventisp=mn


Example21Acubewhosefacesarecoloredissplitinto1000smallcubesofequalsizeThecubesthusobtainedaremixedthoroughlyFindtheprobabilitythatacubedrawnatrandomwillhavetwocoloredfaces

SOLUTIONThetotalnumberofsmallcubesisn=1000Acubehas12edgessothatthereareeightsmallcubeswithtwocoloredfacesoneachedgeHencem=12middot8=96p=mn=0096


Example22 Find the probability that the last two digits of the cube of arandomintegerwillbe11

SOLUTIONRepresentN in the formN =a + 10b + middotmiddotmiddot whereabhelliparearbitrarynumbersrangingfrom0to9ThenN3=a3+30a2b+middotmiddotmiddotFromthiswesee that the last two digits ofN3 are affected only by the values of a and bThereforethenumberofpossiblevaluesisn=100SincethelastdigitofN3isa1 there is one favorable value a = 1Moreover the last digit of (N3 ndash 1)10shouldbe1 ie theproduct3bmustendwitha1Thisoccursonly ifb=7Thusthefavorablevalue(a=1b=7)isuniqueandthereforep=001


Example23FromalotofnitemskaredefectiveFindtheprobabilitythatitemsoutofarandomsampleofsizemselectedforinspectionaredefective

SOLUTIONThenumberofpossiblewaystochoosemitemsoutofnis Thefavorablecasesarethoseinwhichldefectiveitemsamongthekdefectiveitemsareselected(thiscanbedonein ways)andtheremainingmndashlitemsare nondefective ie they are chosen from the total numbernndash k (inways) Thus the number of favorable cases is The required

probabilitywillbe OnecansolveProblems212to220similarly

Example24 Fivepieces aredrawn froma completedomino setFind theprobabilitythatatleastoneofthemwillhavesixdotsmarkedonit

SOLUTIONFindtheprobabilityqofthecomplementaryeventThenp=1ndashqTheprobability thatall fivepieceswillnothaveasix (seeExample23) is

andhence

By a similar passage to the complementary event one can solveProblems221and222

PROBLEMS

21LotteryticketsforatotalofndollarsareonsaleThecostofoneticketisrdollarsandm of all ticketscarryvaluableprizesFind theprobability that asingleticketwillwinavaluableprize

22AdominopieceselectedatrandomisnotadoubleFindtheprobabilitythatthesecondpiecealsoselectedatrandomwillmatchthefirst

23There are four suits in a deck containing 36 cardsOne card is drawnfrom the deck and returned to it The deck is then shuffled thoroughly andanothercardisdrawnFindtheprobabilitythatbothcardsdrawnbelongtothesamesuit

24A letter combination lock contains five disks on a common axisEachdiskisdividedintosixsectorswithdifferentlettersoneachsectorThelockcanopen only if each of the disks occupies a certain positionwith respect to thebody of the lock Find the probability that the lockwill open for an arbitrarycombinationoftheletters

25TheblackandwhitekingsareonthefirstandthirdrowsrespectivelyofachessboardThequeenisplacedatrandominoneof thefreesquaresof thefirst or second row Find the probability that the position for the black kingbecomes checkmate if the positions of the kings are equally probable in anysquaresoftheindicatedrows

26 A wallet contains three quarters and seven dimes One coin is drawn

fromthewalletandthenasecondcoinwhichhappenstobeaquarterFindtheprobabilitythatthefirstcoindrawnisaquarter

27 From a lot containingm defective items andn good ones s items arechosenat randomtobecheckedforqualityAsa resultof this inspectiononefindsthatthefirstkofsitemsaregoodDeterminetheprobabilitythatthenextitemwillbegood

28DeterminetheprobabilitythatarandomlyselectedintegerNgivesasaresult of (a) squaring (b) raising to the fourth power (c) multiplying by anarbitraryintegeranumberendingwitha1

29 On 10 identical cards are written different numbers from 0 to 9Determinetheprobabilitythat(a)atwo-digitnumberformedatrandomwiththegiven cards will be divisible by 18 (b) a random three-digit number will bedivisibleby36

210Findtheprobabilitythattheserialnumberofarandomlychosenbondcontains no identical digits if the serial numbermaybe any five-digit numberstartingwith00001

211TenbooksareplacedatrandomononeshelfFindtheprobabilitythatthreegivenbookswillbeplacedonenexttotheother

212Thenumbers246781112and13arewritten respectivelyoneight indistinguishablecardsTwocardsareselectedat randomfromtheeightFindtheprobabilitythatthefractionformedwiththesetworandomnumbersisreducible

213 Given five segments of lengths 1 3 5 7 and 9 units find theprobabilitythatthreerandomlyselectedsegmentsofthefivewillbethesidesofatriangle

214Twoof10ticketsareprizewinnersFindtheprobabilitythatamongfiveticketstakenatrandom(a)oneisaprizewinner(b)twoareprizewinners(c)atleastoneisaprizewinner

215This is a generalization of Problem 214 There aren+m tickets ofwhichn areprizewinnersSomeonepurchasesk tickets at the same timeFindtheprobabilitythatsoftheseticketsarewinners

216Inalotterythereare90numbersofwhichfivewinByagreementonecanbetanysumonanyoneofthe90numbersoranysetoftwothreefourorfivenumbersWhatistheprobabilityofwinningineachoftheindicatedcases

217Todecreasethetotalnumberofgames2nteamshavebeendividedintotwosubgroupsFind theprobability that the twostrongest teamswillbe (a) indifferentsubgroups(b)inthesamesubgroup

218 A number of n persons are seated in an auditorium that canaccommodate n + k people Find the probability thatm le n given seats are

occupied219 Three cards are drawn at random from a deck of 52 cards Find the

probabilitythatthesethreecardsareathreeasevenandanace220 Three cards are drawn at random from a deck of 36 cards Find the

probabilitythatthesumofpointsofthesecardsis21ifthejackcountsastwopointsthequeenasthreepointsthekingasfourpointstheaceaselevenpointsandtherestassixseveneightnineandtenpoints

221Threeticketsareselectedatrandomfromamongfiveticketsworthonedollar each three tickets worth three dollars each and two tickets worth fivedollars each Find the probability that (a) at least two of them have the sameprice(b)allthreeofthemcostsevendollars

222Thereare2nchildreninlinenearaboxofficewhereticketspricedatanickeleacharesoldWhat is theprobability thatnobodywillhave towait forchangeifbeforeaticketissoldtothefirstcustomerthecashierhas2mnickelsanditisequallyprobablethatthepaymentsforeachticketaremadebyanickelorbyadime

3 GEOMETRICPROBABILITIES

BasicFormulasThegeometricdefinitionofprobabilitycanbeusedonlyiftheprobabilityof

hitting any part of a certain domain is proportional to the size of this domain(lengthareavolumeandsoforth)andisindependentofitspositionandshape

IfthegeometricsizeofthewholedomainequalsS thegeometricsizeofapartofitequalsSBandafavorableeventmeanshittingSBthentheprobabilityofthiseventisdefinedtobe

Thedomainscanhaveanynumberofdimensions


Example31TheaxesofindistinguishableverticalcylindersofradiusrpassthroughanintervallofastraightlineABwhichliesinahorizontalplaneAballofradiusRisthrownatanangleqtothislineFindtheprobabilitythatthisball

willhitonecylinderifanyintersectionpointofthepathdescribedbythecenteroftheballwiththelineABisequallyprobable2

SOLUTIONLetxbethedistancefromthecenteroftheballtothenearestlinethatpassesthroughthecenterofacylinderparalleltothedisplacementdirectionof the center of the ball The possible values of x are determined by theconditions(Figure2)

Thecollisionoftheballwiththecylindermayoccuronlyif0lexleR+rTherequiredprobabilityequalstheratiobetweenthelengthofthesegment

onwhichliethefavorablevaluesofxandthelengthofthesegmentonwhichlieallthevaluesofxConsequently

OnecansolveProblems31to34and324analogously

Example32Ononetrackofamagnetictape200mlongsomeinformationis recorded on an interval of length 20 m and on the second track similarinformationisrecordedEstimatetheprobabilitythatfrom60to85mthereisno interval on the tapewithout recording if the origins of both recordings arelocatedwithequalprobabilityatanypointfrom0to180m

SOLUTIONLetxandybethecoordinatesoforiginoftherecordingswherexgeySince0lexle1800leyle180andxgey the domainof all the possiblevalues ofx and y is a right trianglewith hypotenuse 180m The area of thistriangleisS=12middot1802sqmFindthedomainofvaluesofxandyfavorabletothe given event To obtain a continuous recording it is necessary that theinequalityxndashyle20mholdtrueToobtainarecordingintervallongerthanorequal to 25mwemust have x ndash y ge 5mMoreover to obtain a continuousrecordingontheintervalfrom60to85mwemusthave

FIGURE2

FIGURE3

Drawingtheboundariesoftheindicateddomainswefindthatthefavorablevalues ofx andy are included in a trianglewhose areaSB = 12 middot 152 sqm(Figure3)TherequiredprobabilityequalstheratiooftheareaSB favorable tothegiveneventandtheareaofthedomainScontainingallpossiblevaluesofxandynamely

OnecansolveProblems35to315similarly

Example33 It isequallyprobable that twosignalsreachareceiveratanyinstantofthetimeTThereceiverwillbejammedif thetimedifferenceinthereceptionofthetwosignalsislessthanτFindtheprobabilitythatthereceiverwillbejammed

SOLUTIONLetxandybetheinstantswhenthetwosignalsarereceived

FIGURE4

FIGURE5

ThedomainofallthepossiblevaluesofxyisasquareofareaT2(Figure4)Thereceiverwillbe jammedif |xndashy|leτThegivendomain liesbetween thestraightlinesxndashy=τandxndashy=ndashτItsareaequals

andtherefore


Example 34 Find the probability that the sum of two random positivenumberseachofwhichdoesnotexceedonewillnotexceedoneandthattheirproductwillbeatmost29

SOLUTIONLetxandybethechosennumbersTheirpossiblevaluesare0lexle10leyle1definingintheplaneasquareofareaS=1Thefavorablevaluessatisfytheconditionsx+yle1andxyle29Theboundaryx+y=1dividesthesquareintwosothatthedomainx+yle1representsthelowertriangle(Figure5) The second boundary xy = 29 is a hyperbola The xrsquos of the intersectionpointsoftheseboundariesarex1=13andx2=23Theareaofthefavorabledomainis

Thedesiredprobabilityisp=SBS=0487OnecansolveProblems320to323inasimilarmanner

PROBLEMS

31Abreak occurs at a randompoint on a telephone lineAB of lengthLFindtheprobabilitythatthepointCisatadistancenotlessthanlfromthepointA

32Parallel linesaredrawn inaplaneatalternatingdistancesof15and8

cmEstimatetheprobabilitythatacircleofradius25cmthrownatrandomonthisplanewillnotintersectanyline

33 In a circle of radiusR chords are drawn parallel to a given directionWhat is the probability that the length of a chord selected at randomwill notexceed R if any positions of the intersection points of the chord with thediameterperpendiculartothegivendirectionareequallyprobable

34Infrontofadiskrotatingwithaconstantvelocityweplaceasegmentoflength2h in the plane of the disk so that the line joining themidpoint of thesegment with the center of the disk is perpendicular to this segment At anarbitrary instant a particle flies off the disk Estimate the probability that theparticlewillhitthesegmentifthedistancebetweenthesegmentandthecenterofthediskisl

35ArectangulargridismadeofcylindricaltwigsofradiusrThedistancesbetweentheaxesofthetwigsareaandbrespectivelyFindtheprobabilitythataball of diameterd thrownwithout aimingwill hit the grid in one trial if theflighttrajectoryoftheballisperpendiculartotheplaneofthegrid

36Arectangle3cmtimes5cmisinscribedinanellipsewiththesemi-axesa=100cmandb=10cmsothatitslargersideisparalleltoaFurthermoreoneconstructs fourcirclesofdiameter43cm thatdonot intersect theellipse therectangleandeachother

Determinetheprobabilitythat(a)arandompointwhosepositionisequallyprobable inside theellipsewill turnout tobe insideoneof thecircles (b) thecircleof radius5cmconstructedwith thecenter at thispointwill intersect atleastonesideoftherectangle

37 Sketch five concentric circles of radius kr where k = 1 2 3 4 5respectivelyShadethecircleofradiusrandtwoannuliwiththecorrespondingexteriorradiiof3rand5rThenselectatrandomapointinthecircleofradius5rFindtheprobabilitythatthispointwillbein(a)thecircleofradius2r(b)theshadedregion

38AboatwhichcarriesfreightfromoneshoreofabaytotheothercrossesthebayinonehourWhatistheprobabilitythatashipmovingalongthebaywillbenoticed if theshipcanbeseenfromtheboatat least20minutesbefore theship intersects the direction of the boat and atmost 20minutes after the shipintersects the direction of the boat All times and places for intersection areequallylikely

39 Two points are chosen at random on a segment of length l Find theprobabilitythatthedistancebetweenthepointsislessthanklif0ltkltl

310TwopointsLandMareplacedatrandomonasegmentABoflengthlFindtheprobabilitythatthepointLisclosertoMthantoA

311Onasegmentof length l twopointsareplacedat randomso that thesegmentisdividedintothreepartsFindtheprobabilitythatthesethreepartsofthesegmentaresidesofatriangle

312ThreepointsABCareplacedatrandomonacircleofradiusRWhatistheprobabilitythatthetriangleABCisacute-angled

313 Three line segments each of a length not exceeding l are chosen atrandomWhat is the probability that they can be used to form the sides of atriangle

314TwopointsMandNareplacedonasegmentABoflength lFind theprobabilitythatthelengthofeachofthethreesegmentsthusobtaineddoesnotexceedagivenvaluea(lgeage13)

315AbusoflineAarrivesatastationeveryfourminutesandabusoflineBeverysixminutesThelengthofanintervalbetweenthearrivalofabusoflineA and a bus of lineB may be any number of minutes from zero to four allequallylikely

Findtheprobabilitythat(a)thefirstbusthatarrivesbelongstolineA(b)abusofanylinearriveswithintwominutes

316Two shipsmust arrive at the samemooringsThe timesof arrival forboth ships are independent and equally probable during a given period of 24hours Estimate the probability that one of the shipswill have towait for themooringstobefreeifthemooringtimeforthefirstshipisonehourandforthesecondshiptwohours

317TwopersonshavethesameprobabilityofarrivingatacertainplaceatanyinstantoftheintervalTFindtheprobabilitythatthetimethatapersonhastowaitfortheotherisatmostt

318TwoshipsaresailinginafogonealongabayofwidthLandtheotheracrossthesamebayTheirvelocitiesarev1andv2ThesecondshipemitssoundsthatcanbeheardatadistancedltLFindtheprobabilitythatthesoundswillbeheardonthefirstshipifthetrajectoriesofthetwoshipsmayintersectwithequalprobabilitiesatanypoint

319Abaroflengthl=200mmisbrokenatrandomintopiecesFindtheprobabilitythatatleastonepiecebetweentwobreak-pointsisatmost10mmifthenumberofbreak-pointsis(a)two(b)threeandabreakcanoccurwithequalprobabilityatanypointofthebar

320TwoarbitrarypointsareselectedonthesurfaceofasphereofradiusRWhatistheprobabilitythatanarcofagreatcirclepassingthroughthesepointswillmakeananglelessthanαwhereαltπ

321Asatellitemovesonanorbitbetween60degreesnorthernlatitudeand

60degrees southern latitudeAssuming that the satellitecansplashdownwithequalprobabilityatanypointonthesurfaceoftheearthbetweenthepreviouslymentioned parallels find the probability that the satellite will fall above 30degreesnorthernlatitude

322 A plane is shaded by parallel lines at a distance L between adjacentlines Find the probability that a needle of length l where l lt L thrown atrandomwillintersectsomeline(Buffonrsquosproblem)

323Estimatetheprobabilitythattherootsof(a)thequadraticequationx2+2ax+b=0(b)thecubicequationx3+3ax+2b=0arerealifitisknownthatthe coefficients are equally likely in the rectangle |a| le n |b| lem Find theprobability that under the given conditions the roots of the quadratic equationwillbepositive

324ApointAandthecenterBofacircleofradiusRmoveindependentlyinaplaneThevelocitiesofthesepointsareconstantandequaluandvAtagiveninstantthedistanceABequalsr(rgtR)andtheanglemadebythelineABwiththevectorv equalsβAssuming that all directions for thepointA are equallyprobableestimatetheprobabilitythatthepointAwillbeinsidethecircle

4 CONDITIONALPROBABILITYTHEMULTIPLICATIONTHEOREMFORPROBABILITIES

BasicFormulasTheconditionalprobabilityP(A |B) of the eventA is the probability ofA

under the assumption that the event B has occurred (It is assumed that theprobabilityofB ispositive)TheeventsAandBare independent ifP(A |B)=P(A)Theprobabilityfortheproductoftwoeventsisdefinedbytheformula

whichgeneralizedforaproductofneventsis

TheeventsA1A2hellipAnaresaidtobeindependentifforanymwherem=23hellipnandanykj(j=12hellipn)1lek1ltk2ltmiddotmiddotmiddotltkmlen


Example41ThebreakinanelectriccircuitoccurswhenatleastoneoutofthreeelementsconnectedinseriesisoutoforderComputetheprobabilitythatthebreakinthecircuitwillnotoccurif theelementsmaybeoutoforderwiththerespectiveprobabilities0304and06Howdoestheprobabilitychangeifthefirstelementisneveroutoforder

SOLUTION The required probability equals the probability that all threeelementsareworkingLetAk(k=123)denotetheeventthatthekthelementfunctionsThenp=P(A1A2A3)Sincetheeventsmaybeassumedindependent

Ifthefirstelementisnotoutoforderthen


Example 42 Compute the probability that a randomly selected item is offirstgradeifitisknownthat4percentoftheentireproductionisdefectiveand75percentofthenondefectiveitemssatisfythefirstgraderequirements

ItisgiventhatP(A)=1ndash004=096P(B|A)=075Therequiredprobabilityp=P(AB)=(096)(075)=072SimilarlyonecansolveProblems411to419

Example43Alotof100itemsundergoesaselectiveinspectionTheentirelotisrejectedifthereisatleastonedefectiveiteminfiveitemscheckedWhatisthe probability that the given lot will be rejected if it contains 5 per centdefectiveitems

SOLUTIONFindtheprobabilityqofthecomplementaryeventAconsistingofthesituationinwhichthelotwillbeacceptedThegiveneventisanintersectionof fiveeventsA=A1A2A3A4A5whereAk(k=12345)means that thekthitemcheckedisgood

Theprobabilityof theeventA1 isP(A1)=95100 since thereareonly100

itemsofwhich95aregoodAftertheoccurrenceoftheeventA1thereremain99itemsofwhich94are

good and thereforeP(A2 |A1) = 9499 AnalogouslyP(A3 |A1A2) = 9398P(A4|A1A2A3)=9297andP(A5|A1A2A3A4)=9196Accordingtothegeneralformulawefindthat

Therequiredprobabilityp=1ndashq=023OnecansolveProblems420to435similarly

PROBLEMS

41Twomarksmenwhose probabilities of hitting a target are 07 and 08respectivelyfireoneshoteachFindtheprobabilitythatatleastoneofthemwillhitthetarget

42Theprobability that thekthunitofacomputer isoutoforderduringatimeT equals pk (k= 1 2hellip n) Find the probability that during the givenintervaloftimeatleastoneofnunitsofthiscomputerwillbeoutoforderifalltheunitsrunindependently

43Theprobabilityoftheoccurrenceofaneventineachperformanceofanexperiment is02Theexperimentsarecarriedoutsuccessivelyuntil thegiveneventoccursFind theprobability that itwillbenecessary toperforma fourthexperiment

44Theprobabilitythatanitemmadeonthefirstmachineisoffirstgradeis07Theprobability that an itemmadeon the secondmachine is firstgrade is08 The first machinemakes two items and the secondmachine three itemsFindtheprobabilitythatallitemsmadewillbeoffirstgrade

45Abreak in an electric circuitmay occur only if one elementK or twoindependent elementsK1 andK2 are out of orderwith respective probabilities0302and02Findtheprobabilityofabreakinthecircuit

46AdevicestopsasaresultofdamagetoonetubeofatotalofNTolocatethis tube one successively replaces each tube with a new one Find theprobabilitythatitwillbenecessarytocheckntubesiftheprobabilityispthatatubewillbeoutoforder

47Howmanynumbersshouldbeselectedfromatableofrandomnumberssothattheprobabilityoffindingatleastoneevennumberamongthemis09

48TheprobabilitythatasaresultoffourindependenttrialstheeventAwilloccurat leastonce is05Find theprobability that theeventwilloccur inonetrialifthisprobabilityisconstantthroughalltheothertrials

49An equilateral triangle is inscribed in a circle of radiusRWhat is theprobability that four points taken at random in the given circle are inside thistriangle

410Findtheprobabilitythatarandomlywrittenfractionwillbeirreducible(Chebyshevrsquosproblem)3

411 If twomutually exclusive eventsA andB are such thatP(A)ne0andP(B)ne0aretheseeventsindependent

412 The probability that the voltage of an electric circuitwill exceed theratedvalueisp1Foranincreaseinthevoltage theprobability thatanelectricdevicewillstopisp2Findtheprobabilitythatthedevicewillstopasaresultofanincreaseinthevoltage

413Amotorcyclistinaracemustpassthrough12obstaclesplacedalongacourse AB he will stop at each of them with probability 01 Knowing theprobability07withwhich themotorcyclistpasses fromB to the final pointCwithoutstopsfindtheprobabilitythatnostopswilloccuronthesegmentAC

414 Three persons play a game under the following conditions At thebeginning thesecondand thirdplay in turnsagainst the first In thiscase thefirstplayerdoesnotwin(butmightnotloseeither)andtheprobabilitiesthatthesecondandthirdwinareboth03Ifthefirstdoesnotlosehethenmakesonemoveagainsteachoftheothertwoplayersandwinsfromeachofthemwiththeprobability 04 After this the game ends Find the probability that the firstplayerwinsfromatleastoneoftheothertwo

415Amarksmanhitsatargetwiththeprobability23IfhescoresahitonthefirstshotheisallowedtofireanothershotatanothertargetTheprobabilityoffailingtohitbothtargetsinthreetrialsis05Findtheprobabilityoffailingtohitthesecondtarget

416 Some items are made by two technological procedures In the firstprocedure an item passes through three technical operations and theprobabilitiesofadefectoccurringintheseoperationsare0102and03Inthesecond procedure there are two operations and the probability of a defectoccurringineachofthemis03Determinewhichtechnologyensuresagreaterprobability of first grade production if in the first case for a good item theprobabilityoffirstgradeproductionis09andinthesecondcase08

417 The probabilities that an item will be defective as a result of amechanicalanda thermalprocessarep1andp2 respectivelyTheprobabilities

ofeliminatingdefectsarep3andp4respectivelyFind(a)howmanyitemsshouldbeselectedafterthemechanicalprocessin

order to be able to claim that at least one of them can undergo the thermalprocesswithachanceofeliminatingthedefect(b)theprobabilitythatat leastone of three itemswill have a nonremovable defect after passing through themechanicalandthermalprocesses

418 Show that if the conditional probability P(A | B) exceeds theunconditionalprobabilityP(A)thentheconditionalprobabilityP(B|A)exceedstheunconditionalprobabilityP(B)

419Atargetconsistsoftwoconcentriccirclesofradiuskrandnrwherekltn If it is equally probable that one hits any part of the circle of radius nrestimatetheprobabilityofhittingthecircleofradiuskrintwotrials

420With six cards eachcontainingone letterone forms theword latentThecardsarethenshuffledandatrandomcardsaredrawnoneatatimeWhatistheprobabilitythatthearrangementofletterswillformthewordtalent

421AmanhasforgottenthelastdigitofatelephonenumberandthereforehedialsitatrandomFindtheprobabilitythathemustdialatmostthreetimesHow does the probability change if one knows that the last digit is an oddnumber

422Somem lottery ticketsoutofa totalofnare thewinnersWhat is theprobabilityofawinnerinkpurchasedtickets

423Threelotteryticketsoutofatotalof40000arethebigprizewinnersFind(a)theprobabilityofgettingatleastonebigprizewinner(ticket)per1000tickets(b)howmanyticketsshouldbepurchasedsothattheprobabilityofonebigwinnerisatleast05

424 Six regular drawings of state bonds plus one supplementary drawingafter the fifth regular one take place annually From a total of 100 000 serialnumbers the winners are 170 in each regular drawing and 270 in eachsupplementaryoneFind theprobability thatabondwinsafter tenyears in (a)anydrawing(b)asupplementarydrawing(c)aregulardrawing

425 Consider four defective items one item has the paint damaged thesecond has a dent the third is notched and the fourth has all three defectsmentionedConsideralsotheeventA that thefirst itemselectedatrandomhasthepaintdamagedtheeventBthattheseconditemhasadentandtheeventCthatthethirditemisnotchedArethegiveneventsindependentinpairsorasawholeset

426LetA1A2hellipAnbeasetofeventsindependentinpairsIsittruethattheconditionalprobabilitythataneventoccurscomputedundertheassumption

thatothereventsofthesamesethaveoccurredistheunconditionalprobabilityofthisevent

427AsquareisdividedbyhorizontallinesintonequalstripsThenapointwhose positions are equally probable in the strip is taken in each strip In thesamewayonedrawsnndash1verticallinesFindtheprobabilitythateachverticalstripwillcontainonlyonepoint

428 A dinner party of 2n persons has the same number of males andfemalesFindtheprobabilitythattwopersonsofthesamesexwillnotbeseatednexttoeachother

429Apartyconsistingof fivemalesand10 females isdividedat randomintofivegroupsofthreepersonseachFindtheprobabilitythateachgroupwillhaveonemalemember

430Anurncontainsn+midenticalballsofwhichnarewhiteandmblackwheremgenApersondrawsballsntimestwoballsatatimewithoutreturningthemtotheurnFindtheprobabilityofdrawingapairofballsofdifferentcolorseachtime

431Anurncontainsnballsnumberedfrom1tonTheballsaredrawnoneata timewithoutbeing replaced in theurnWhat is theprobability that in thefirst k draws the numbers on the balls will coincide with the numbers of thedraws

432AnurncontainstwokindsofballswhiteonesandblackonesTheballsaredrawnoneatatimeuntilablackballappearsandeachtimewhenawhiteballisdrawnitisreturnedtotheurntogetherwithtwoadditionalballsFindtheprobabilitythatinthefirst50trialsnoblackballswillbedrawn

433Therearen+mmen in line for tickets that arepricedat fivedollarseachnof thesemenhave five-dollarbillsandmwheremlen+1have ten-dollarbillsEachpersonbuysonlyoneticketThecashierhasnomoneybeforetheboxofficeopensWhatistheprobabilitythatnooneinthelinewillhavetowaitforchange

434Theproblemisthesameasin433butnowtheticketcostsonedollarandn of the customers have one-dollar billswhereasm have five-dollar billswhere2mlen+1

435OftwocandidatesNo1receivesnvoteswhereasNo2receivesm(ngtm)votesEstimatetheprobabilitythatatalltimesduringthevotecountNo1willleadNo2

5 THEADDITIONTHEOREMFORPROBABILITIES

BasicFormulasTheprobabilityoftheunionoftwoeventsisgivenby

whichcanbeextendedtoaunionofanynumberofevents

Formutuallyexclusiveeventstheprobabilityofaunionofeventsisthesumoftheprobabilitiesoftheseeventsthatis


Example51Findtheprobabilitythatalotof100itemsofwhichfivearedefectivewill be accepted in a test of a randomly selected sample containinghalfthelotiftobeacceptedthenumberofdefectiveitemsinalotof50cannotexceedone

SOLUTIONLetAbetheeventdenotingthatthereisnodefectiveitemamongthosetestedandBthatthereisonlyonedefectiveitemTherequiredprobabilityisp=P(A)+P(B)TheeventsAandBaremutuallyexclusiveThusp=P(A)+P(B)

There are ways of selecting 50 items from a total of 100 From 95nondefective items one can select 50 items in ways ThereforeP(A) =

Analogously Then

Problems51to512aresolvedsimilarly

Example52TheschemeoftheelectriccircuitbetweentwopointsMandN

is given in Figure 6 Malfunctions during an interval of time T of differentelements of the circuit represent independent events with the followingprobabilities(Table1)

TABLE1

Findtheprobabilityofabreakinthecircuitduringtheindicatedintervaloftime

SOLUTIONDenotebyAj(j=12)theeventmeaningthatanelementKjisoutoforderbyAthatatleastoneelementKjisoutoforderandbyBthatallthreeelementsLi(i=123)areoutoforderThentherequiredprobabilityis

Since

wegetp 085OnecansolveProblems513to516analogously

Example53TheoccurrenceoftheeventAisequallyprobableatanyinstantof the intervalT The probability thatA occurs during this interval is p It isknown that during an interval tltT the given event does not occur Find theprobabilityPthattheeventAwilloccurduringtheremainingintervaloftime

FIGURE6

SOLUTIONTheprobabilitypthattheeventAoccursduringtheintervalT is

theprobability thatthegiveneventoccursduringtimetplustheproductof

theprobability thatA will not occur during t by the conditionalprobabilitythatitwilloccurduringtheremainingtimeifitdidnotoccurbeforeThusthefollowingequalityholdstrue

Fromthiswefind

Example54Anurn containsnwhite ballsm blackballs and l red ballswhich are drawn at random one at a time (a) without replacement (b) withreplacementofeachballtotheurnaftereachdrawFindtheprobabilitythatinbothcasesawhiteballwillbedrawnbeforeablackone

SOLUTIONLetPI be the probability for awhite ball to be drawn before ablackoneandPIIbetheprobabilityforablackballtobedrawnbeforeawhiteball

The probability PI is the sum of probabilities of drawing a white ballimmediately after a red ball two red balls and so forth Thus in the casewithoutreplacementwehave

andinthecasewithreplacement

To obtain the probabilities PII replace n bym andm by n in the precedingformulasFromthisitfollowsinbothcasesthatPIPII=nmFurthermoresincePI+PII=1therequiredprobabilityinthecasewithoutreplacementisalsoPI=n(n+m)


Example55Apersonwroten letterssealed theminenvelopesandwrotethe different addresses randomly on each of themFind the probability that atleastoneoftheenvelopeshasthecorrectaddress

SOLUTION Let the event Ak mean that the kth envelope has the correct

addresswherek=12hellipnThedesiredprobabilityis TheeventsAkaresimultaneousforanykjihellipthefollowingequalitiesobtain

andfinally

Usingtheformulafortheprobabilityofasumofneventsweobtain

or

Forlargenpasymp1ndashendash1SimilarlyonecansolveProblems532to538

PROBLEMS

51 Any one of four mutually exclusive events may occur with thecorrespondingprobabilities001200100006and0002Find theprobabilitythattheoutcomeofanexperimentisatleastoneoftheseevents

52Amarksmanfiresoneshotata targetconsistingofacentralcircleandtwoconcentricannuliTheprobabilitiesofhitting thecircleand theannuliare020015and010respectivelyFindtheprobabilityofnothittingthetarget

53Aballisthrownatasquaredividedinton2identicalsquaresTheprobability that theballwillhitasmallsquareof thehorizontalstrip i

andverticalstrip j ispij Find theprobability that theballwillhitahorizontalstrip

54Twoidenticalcoinsof radiusrareplaced insideacircleof radiusRatwhichapointisthrownatrandomFindtheprobabilitythatthispointhitsoneofthecoinsifthecoinsdonotoverlap

55What is theprobabilityofdrawingfromadeckof52cardsafacecard(jackqueenorking)ofanysuitoraqueenofspades

56Aboxcontainsten20-centstampsfive15-centstampsandtwo10-centstampsOnedrawssixstampsatrandomWhatistheprobabilitythattheirsumdoesnotexceedonedollar(100cents)

57GiventheprobabilitiesoftheeventsAandABfindtheprobabilityoftheeventA

58Provethatfromthecondition

itfollowsthattheeventsAandBareindependent59TheeventBincludestheeventAProvethatP(A)leP(B)510 Two urns contain balls differing only in color The first urn has five

white11blackandeightredballsthesecondhas10whiteeightblackandsixred ballsOne ball at a time is drawn at random from both urnsWhat is theprobabilitythatbothballswillbeofthesamecolor

511Twoparallelstrips10mmwidearedrawnintheplaneatadistanceof155mmAlongaperpendiculartothesestripsatadistanceof120mmliethecentersofcirclesof radius10mmFind theprobability thatat leastonecirclewillcrossoneofthestripsifthecentersofthecirclesaresituatedalongthelineindependentofthepositionofthestrips

512 The seeds of n plants are sown in a line along the road at equaldistancesfromeachotherTheprobabilitythatapedestriancrossingtheroadatanypointwilldamageoneplantisp(pltln)Findtheprobabilitythatthemthpedestrianwhocrossestheroadatanonpredeterminedpointwilldamageaplantifthepedestrianscrosstheroadsuccessivelyandindependently

513 Find the probability that a positive integer randomly selectedwill benondivisibleby(a)twoandthree(b)twoorthree

514Theprobabilityofpurchasingaticketinwhichthesumsofthefirstandlastthreedigitsareequalis005525Whatistheprobabilityofreceivingsuchaticketamongtwoticketsselectedatrandomifbothtickets(a)haveconsecutivenumbers(b)areindependentofeachother

515ProvethatifP(A)=aandP(B)=bthen

516GiventhatP(Xle10)=09P(|Y|le1)=095provethatregardlessoftheindependenceofXandYifZ=X+Ythenthefollowinginequalitieshold

517AgamebetweenAandB isconductedunder thefollowingrulesasaresultofthefirstmovealwaysmadebyAhecanwinwiththeprobability03ifAdoesnotwininthefirstmoveBplaysnextandcanwinwiththeprobability05ifinthismoveBdoesnotwinAmakesthenextmoveinwhichhecanwinwiththeprobability04FindtheprobabilitiesofwinningforAandB

518Giventheprobabilitypthatacertainsportsmanimproveshispreviousscoreinonetrialfindtheprobabilitythatthesportsmanwillimprovehisscoreinacompetitioninwhichtwotrialsareallowed

519 Player A plays two games each in turn with players B andC TheprobabilitiesthatthefirstgameiswonbyBandCare01and02respectively

theprobabilitythatthesecondgameiswonbyBis03andbyC04Findtheprobabilitythat(a)Bwinsfirst(b)Cwinsfirst520 From an urn containing n balls numbered from 1 to n two balls are

drawnsuccessively thefirstball is returned to theurn if itsnumber is1Findtheprobabilitythattheballnumbered2isdrawnonthesecondtrial

521 PlayerA plays in turn with playersB andC with the probability ofwinning in each game 025 he ends the game after the first loss or after twogamesplayedwitheachoftheotherplayersFindtheprobabilitiesthatBandCwin

522The probability that a device breaks after it has been used k times isG(k)Findtheprobabilitythatthedeviceisoutoforderafternconsecutiveusesifduringthepreviousmoperationsitwasnotoutoforder

523TwopersonsalternatelyflipacoinTheonewhogetsheadsfirstisthewinnerFindtheprobabilitiesofwinningforeachplayer

524ThreepersonssuccessivelytossacoinTheonewhogetsheadsfirstisthewinnerFindtheprobabilitiesofwinningforeachplayer

525 The probability of gaining a point without losing service in a gamebetween two evenlymatchedvolleyball teams is 05 Find the probability thattheservingteamwillgainapoint

526Anurn containsnwhite andm black ballsTwoplayers successivelydraw one ball at a time and each time return the ball to the urn The gamecontinues until one of them draws a white ball Find the probability that thewhiteballwillbefirstdrawnbytheplayerwhostartsthegame

527 Two marksmen shoot in turn until one of them hits the target Theprobability of hitting the target is 02 for the first marksman and 03 for thesecondoneFindtheprobabilitythatthefirstmarksmanfiresmoreshotsthanthesecond

528Provethevalidityoftheequality

529 Simplify the general formula for the probability of a union of eventsapplicable to the casewhen theprobabilities forproductsof equalnumbersofeventscoincide

530Provethat

531 Prove that for any eventsAk (k = 0 1hellipn) the following equalityholdstrue

532Anurncontainsnballsnumberedfrom1tonTheballsaredrawnfromtheurnoneatatimewithoutreplacementFindtheprobabilitythatinsomedrawthenumberontheballcoincideswiththenumberofthetrial

533Anauditoriumhasnnumberedseatsnticketsaredistributedamongnpersons What is the probability that m persons will be seated at seats thatcorrespondtotheirticketnumbersifalltheseatsareoccupiedatrandom

534Atrainconsistsofncarsk(kgen)passengersgetonitandselecttheircarsatrandomFindtheprobabilitythattherewillbeatleastonepassengerineachcar

535Twopersonsplayuntil there is avictorywhichoccurswhen the firstwinsmgamesorthesecondngamesTheprobabilitythatagameiswonispforthefirstplayerandq=1ndashpforthesecondFindtheprobabilitythatthewholecompetitioniswonbythefirstplayer

536Two persons have agreed that a prizewill go to the onewhowins agivennumberof gamesThegame is interruptedwhenm games remain to bewonbythefirstplayerandnbythesecondHowshouldthestakesbedividediftheprobabilityofwinningagameis05foreachplayer

537ThisistheproblemoffourliarsOneperson(a)outoffourabcanddreceivesinformationthathetransmitsintheformofaldquoyesrdquoorldquonordquosignaltothesecondperson(b)Thesecondpersontransmitstothethird(c)thethirdtothefourth (d) and the fourth communicates the received information in the samemanner as all theothersGiven the fact thatonlyoneperson in three tells thetruth find the probability that the first liar tells the truth if the fourth told thetruth

538SomeparallellinesseparatedbythedistanceLaredrawninahorizontalplaneAconvexcontourofperimeters is randomly thrownat thisplaneFindthe probability that it will intersect one of the parallels if the diameter of thesmallestcirclecircumscribedaboutthecontourislessthanL

6 THETOTALPROBABILITYFORMULA

BasicFormulasTheprobabilityP(A) thataneventAwilloccursimultaneouslywithoneof

theeventsH1H2hellipHn formingacompletesetofmutuallyexclusiveevents(hypotheses)isgivenbythetotalprobabilityformula

where


Example 61 Among n personsm le n prizes are distributed by randomdrawing in turn from a box containing n tickets Are the chances of winningequalforallparticipantsWhenisitbesttodrawaticket

SOLUTIONDenotebyAktheeventthatconsistsofdrawingawinningticketinkdrawsfromtheboxAccordingtotheresultsoftheprecedingexperimentsonecanmakek+1hypothesesLetthehypothesisHksmeanthatamongkdrawnticketssareprizewinnersTheprobabilitiesofthesehypothesesare

where

Sincetherearenndashkticketsleftofwhichmndashsarewinnersformges

Bythetotalprobabilityformulawefind

where =0forsgtmThisequalitycanalsobewrittenintheform

Wehave

thatisthefollowingequalityholdstrue

TherequiredprobabilityP(Ak)=mnforanykThereforeallparticipantshaveequalchancesandthesequenceinwhichtheticketsaredrawnisnotimportant

AnalogouslyonecansolveProblems61to617

Example62Amarkedball canbe in the firstor secondof twournswithprobabilitiespand1ndashpTheprobabilityofdrawing themarkedball fromtheurn inwhich it is located isP(Pne1)What is thebestway tousendrawsofballsfromanyurnsothattheprobabilityofdrawingthemarkedballislargestiftheballisreturnedtoitsurnaftereachdraw

SOLUTIONDenotebyAtheeventconsistingofdrawingthemarkedballThehypothesesareH1thattheballisinthefirsturnH2thattheballisinthesecondurnByassumptionP(H1)=pP(H2)=1ndashpIfmballsaredrawnfromthefirsturnandnndashmballsfromthesecondurntheconditionalprobabilitiesofdrawingthemarkedballare

Accordingtothetotalprobabilityformulatherequiredprobabilityis

One should findm so that the probabilityP(A) is largestDifferentiatingP(A)with respect tom (to find an approximate value ofmwe assume thatm is acontinuousvariable)weobtain

SettingdP(A)dm=0wegettheequality(1ndashP)2mndashn=(1ndashp)pThus

TheprecedingformulaisusedinsolvingProblems618and619

PROBLEMS

61Therearetwobatchesof10and12itemseachandonedefectiveitemineach batchAn item taken at random from the first batch is transferred to thesecondafterwhichoneitemistakenatrandomfromthesecondbatchFindtheprobabilityofdrawingadefectiveitemfromthesecondbatch

62TwodominopiecesarechosenatrandomfromacompletesetFindtheprobabilitythatthesecondpiecewillmatchthefirst

63Twournscontainrespectivelym1andm2whiteballsandn1andn2blackballsOneballisdrawnatrandomfromeachurnandthenfromthetwodrawnballsoneistakenatrandomWhatistheprobabilitythatthisballwillbewhite

64TherearenurnseachcontainingmwhiteandkblackballsOneballisdrawnfromthefirsturnandtransferredtothesecondurnThenoneballistakenatrandomfrsmthesecondurnandtransferredtothethirdandsoonFindtheprobabilityofdrawingawhiteballfromthelasturn

65Therearefivegunsthatwhenproperlyaimedandfiredhaverespectiveprobabilitiesofhittingthetargetasfollows05060708and09OneofthegunsischosenatrandomaimedandfiredWhatistheprobabilitythatthetargetishit

66ForqualitycontrolonaproductionlineoneitemischosenforinspectionfromeachofthreebatchesWhatistheprobabilitythatfaultyproductionwillbedetectedifinoneofthebatches23oftheitemsarefaultyandintheothertwotheyareallgood

67 A vacuum tube may come from any one of three batches withprobabilitiesp1p2andp3wherep1=p3=025andp2=05Theprobabilitiesthatavacuumtubewilloperateproperlyforagivennumberofhoursareequalto 01 02 and 04 respectively for these batches Find the probability that arandomlychosenvacuumtubewilloperateforthegivennumberofhours

68PlayerA plays twoopponents alternatelyTheprobability that hewinsfromoneatthefirsttrialis05andtheprobabilitythathewinsfromtheotherat

thefirsttrialis06Theseprobabilitiesincreaseby01eachtimetheopponentsrepeat the play againstA Assume thatA wins the first two games Find theprobabilitythatAwilllosethethirdgameifhisopponentinthefirstgameisnotknownandiftiesareexcluded

69Aparticularmaterialusedinaproductionprocessmaycomefromoneofsixmutuallyexclusivecategorieswithprobabilities009016025025016and009Theprobabilitiesthatanitemofproductionwillbeacceptableifitismadefrommaterials in thesecategoriesarerespectively0203040403and02Findtheprobabilityofproducinganacceptableitem

610 An insulating plate 100 mm long covers two strips passingperpendicular to its length Their boundaries are located respectively at thedistancesof2040mmand6590mmfromtheedgeoftheplateAholeof10mm diameter ismade so that its center is located equiprobably on the plateFindtheprobabilityofanelectriccontactwithanyofthestripsifaconductorisapplied fromabove toanarbitrarypoint locatedat thesamedistance from thebaseoftheplateasthecenterofthehole

611TheprobabilitythatkcallsarereceivedatatelephonestationduringanintervaloftimetisequaltoPt(k)AssumingthatthenumbersofcallsduringtwoadjacentintervalsareindependentfindtheprobabilityP2t(S)thatscallswillbereceivedduringaninterval2t

612Findtheprobabilitythat100lightbulbsselectedatrandomfromalotof1000willbenondefectiveifanynumberofdefectivebulbsfrom0to5per1000isequallyprobable

613 A white ball is dropped into a box containing n balls What is theprobabilityofdrawing thewhiteball from thisbox ifall thehypothesesabouttheinitialcolorcompositionoftheballsareequallyprobable

614Inaboxare15tennisballsofwhichninearenewForthefirstgamethreeballsareselectedat randomandafterplay theyare returned to theboxForthesecondgamethreeballsarealsoselectedatrandomFindtheprobabilitythatalltheballstakenforthesecondgamewillbenew

615Therearethreequartersandfournickels intherightpocketofacoatandsixquartersandthreenickelsintheleftpocketFivecoinstakenatrandomfrom the rightpocketare transferred to the leftpocketFind theprobabilityofdrawing a quarter at random from the left pocket after this transfer has beenmade

616AnexaminationisconductedasfollowsThirtydifferentquestionsareenteredinpairson15cardsAstudentdrawsonecardatrandomIfhecorrectlyanswers both questions on the drawn card he passes If he correctly answers

onlyonequestionon thedrawncardhedrawsanother cardand theexaminerspecifieswhichofthetwoquestionsonthesecondcardistobeansweredIfthestudent correctly answers the specified question he passes In all othercircumstanceshefails

If the student knows the answers to 25 of the questions what is theprobabilitythathewillpasstheexamination

617Underwhatconditionsdoesthefollowingequalityhold

618Oneoftwournseachcontaining10ballshasamarkedballAplayerhas the right todraw successively20balls fromeitherof theurns each timereturning the ball drawn to the urn How should one play the game if theprobabilitythatthemarkedballisinthefirsturnis23Findthisprobability

619Ten helicopters are assigned to search for a lost airplane each of thehelicopters can be used in one out of two possible regionswhere the airplanemight be with the probabilities 08 and 02 How should one distribute thehelicopters so that the probability of finding the airplane is the largest if eachhelicoptercanfindthelostplanewithinitsregionofsearchwiththeprobability02 and each helicopter searches independentlyDetermine the probability offindingtheplaneunderoptimalsearchconditions

7 COMPUTATIONOFTHEPROBABILITIESOFHYPOTHESESAFTERATRIAL(BAYESrsquoFORMULA)

BasicFormulasTheprobabilityP(Hk |A)ofthehypothesisHkaftertheeventAoccurredis

givenbytheformula

where

andthehypothesesHj(j=1hellipn)formacompletesetofmutuallyexclusiveevents


Example71AtelegraphiccommunicationssystemtransmitsthesignalsdotanddashAssumethatthestatisticalpropertiesoftheobstaclesaresuchthatanaverageof25of thedotsand13of thedashesarechangedSuppose that theratiobetweenthetransmitteddotsandthetransmitteddashesis53Whatistheprobabilitythatareceivedsignalwillbethesameasthetransmittedsignalif(a)thereceivedsignalisadot(b)thereceivedsignalisadash

SOLUTION LetA be the event that a dot is received andB that a dash isreceived

OnecanmaketwohypothesesH1thatthetransmittedsignalwasadotandH2 that the transmitted signalwas a dashBy assumptionP(H1)P(H2)=53MoreoverP(H1)+P(H2)=1ThereforeP(H1)=58P(H2)=38Oneknowsthat

The probabilities of A and B are determined from the total probabilityformula

Therequiredprobabilitiesare


Example72Therearetwolotsofitemsitisknownthatalltheitemsofonelot satisfy the technical standards and 14 of the items of the other lot aredefective Suppose that an item from a lot selected at random turns out to begoodFindtheprobabilitythataseconditemofthesamelotwillbedefectiveifthefirstitemisreturnedtothelotafterithasbeenchecked

SOLUTIONConsiderthehypothesesH1thatthelotwithdefectiveitemswasselectedandH2thatthelotwithnondefectiveitemswasselectedLetAdenotetheevent that the first item isnondefectiveBy theassumptionof theproblemP(H1)=P(H2)=12P(A|H1)=34P(A|H2)=1Thususingtheformulaforthe totalprobabilitywefind that theprobabilityof theeventAwillbeP(A)=l2[(34)+1]=78Afterthefirsttrial theprobabilitythatthelotwillcontaindefectiveitemsis

Theprobabilitythatthelotwillcontainonlygooditemsisgivenby

LetB be the event that the item selected in the first trial turns out to be

defectiveTheprobabilityofthiseventcanalsobefoundfromtheformulaforthetotalprobabilityIfp1andp2aretheprobabilitiesofthehypothesesH1andH2afteratrialthenaccordingtotheprecedingcomputationsp1=37p2=47FurthermoreP(B|H1)=14P(B|H2)=0ThereforetherequiredprobabilityisP(B)=(37)middot(14)=328

OnecansolveProblems717and718similarly

PROBLEMS

71 Consider 10 urns identical in appearance of which nine contain twoblackandtwowhiteballseachandonecontainsfivewhiteandoneblackballAnurnispickedatrandomandaballdrawnatrandomfromitiswhiteWhatistheprobabilitythattheballisdrawnfromtheurncontainingfivewhiteballs

72Assumethatk1urnscontainmwhiteandnblackballseachandthatk2urns containm white and n black balls each A ball drawn from a randomlyselectedurnturnsouttobewhiteWhatistheprobabilitythatthegivenballwillbedrawnfromanurnofthefirsttype

73 Assume that 96 per cent of total production satisfies the standardrequirementsAsimplifiedinspectionschemeacceptsastandardproductionwiththeprobability098 andanonstandardonewith theprobability005Find theprobability that an item undergoing this simplified inspection will satisfy thestandardrequirements

74Fromalotcontainingfiveitemsoneitemisselectedwhichturnsouttobe defective Any number of defective items is equally probable Whathypothesisaboutthenumberofdefectiveitemsismostprobable

75Findtheprobabilitythatamong1000lightbulbsnonearedefectiveifallthebulbsofarandomlychosensampleof100bulbsturnouttobegoodAssumethat anynumberofdefective light bulbs from0 to5 in a lot of1000bulbs isequallyprobable

76ConsiderthatDplaysagainstanunknownadversaryunderthefollowingconditionsthegamecannotendinatiethefirstmoveismadebytheadversaryin case he loses the nextmove ismade byDwhose gainmeanswinning thegameifDlosesthegameisrepeatedunderthesameconditionsBetweentwoequallyprobableadversariesBandCB has theprobability04ofwinning inthefirstmoveand03inthesecondChastheprobability08ofwinninginthefirstmoveand06inthesecondDhastheprobability03ofwinninginthefirstmoveregardlessoftheadversaryandrespectively0507whenplayingagainstBandCinthesecondmoveThegameiswonbyD

Whatistheprobabilitythat(a)theadversaryisB(b)theadversaryisC77Consider 18marksmen ofwhom five hit a targetwith the probability

08sevenwiththeprobability07fourwiththeprobability06andtwowiththeprobability05A randomlyselectedmarksman firesa shotwithouthitting thetargetTowhatgroupisitmostprobablethathebelongs

78Theprobabilities that threepersonshita targetwithadartareequal to45 34 and 23 In a simultaneous throw by all three marksmen there areexactlytwohitsFindtheprobabilitythatthethirdmarksmanwillfail

79ThreehuntersshootsimultaneouslyatawildboarwhichiskilledbyonebulletFindtheprobabilitythattheboariskilledbythefirstsecondorthethirdhunteriftheprobabilitiesoftheirhittingtheboararerespectively0204and06

710Adart thrownat randomcanhitwithequalprobabilityanypointofaregionSthatconsistsoffourpartsrepresenting50percent30percent12percentand8percentoftheentireregionWhichpartofregionSismostlikelytobehit

711InanurntherearenballswhosecolorsarewhiteorblackwithequalprobabilitiesOne draws k balls from the urn successivelywith replacementWhatistheprobabilitythattheurncontainsonlywhiteballsifnoblackballsaredrawn

712Thefirstbornofasetoftwinsisaboywhatistheprobabilitythattheotherisalsoaboyifamongtwinstheprobabilitiesoftwoboysortwogirlsareaandbrespectivelyandamongtwinsofdifferentsexestheprobabilitiesofbeingbornfirstareequalforbothsexes

713Consideringthattheprobabilityofthebirthoftwinsofthesamesexistwicethatoftwinsofdifferentsexesthattheprobabilitiesoftwinsofdifferentsexesareequal inanysuccessionand that theprobabilitiesofaboyandagirlare respectively 051 and 049 find the probability of a second boy if thefirstbornisaboy

714 Two marksmen fire successively at a target Their probabilities ofhittingthetargetonthefirstshotsare04and05andtheprobabilitiesofhittingthe target in the next shots increase by 005 for each of them What is theprobabilitythatthefirstshotwasfiredbythefirstmarksmanifthetargetishitbythefifthshot

715ConsiderthreeindependenttrialsinwhichtheeventAoccurswiththeprobability02TheprobabilityoftheoccurrenceoftheeventBdependsonthenumberofoccurrencesofAIftheeventAoccursoncethisprobabilityis01ifAoccurstwiceitis03ifAoccursthreetimesitis07iftheeventAdoesnotoccurtheeventBisimpossibleFindthemostprobablenumberofoccurrences

ofAifitisknownthatBhasoccurred716TherearenstudentsinatechnicalschoolOfthesenkwherek=12

3areintheirsecondyearTwostudentsarerandomlyselectedoneofthemhasbeen studying formore years than the otherWhat is the probability that thisstudenthasbeenstudyingforthreeyears

717 The third item of one of three lots of items is of second grade theremainingitemsareoffirstgradeAnitemselectedfromoneofthelotsturnsouttobeoffirstgradeFindtheprobabilitythatitwastakenfromthelotcontainingsecond grade items Find the same probability under the assumption that aseconditemselectedfromthesamelotturnsouttobeoffirstgradeifthefirstitemisreturnedtothelotafterinspection

718ConsideralotofeightitemsofonesampleFromthedataobtainedbycheckingone-halfofthelotthreeitemsturnouttobetechnicallygoodandoneisdefectiveWhatistheprobabilitythatincheckingthreesuccessiveitemsonewill turnout tobegoodand twodefective ifanynumberofdefective items isequallyprobableinthegivenlot

8 EVALUATIONOFPROBABILITIESOFOCCURRENCEOFANEVENTINREPEATEDINDEPENDENTTRIALS

BasicFormulasTheprobabilityPnmthataneventoccursmtimesinnindependenttrialsin

which theprobabilityofoccurrenceof theevent isp isgivenby thebinomialdistributionformula

whereq=1ndashpTheprobabilityforrealizationoftheeventatleastmtimesinntrialscanbe

computedfromtheformula

Theprobabilityofoccurrenceoftheeventatleastonceinntrialswillbe

ThenumberoftrialsthatmustbecarriedoutinordertoclaimthatagiveneventoccursatleastoncewithaprobabilityatleastPisgivenbytheformula

wherepistheprobabilityofoccurrenceoftheeventineachofthetrialsThemostprobablevaluemicroof thenumbermofoccurrencesof theeventA

equalstheintegralpartofthenumber(n+1)pandif(n+1)pisanintegerthelargestvalueoftheprobabilityisattainedfortwonumbersmicro1=(n+1)pndash1andmicro2=(n+1)p

IfthetrialsareindependentbuttheprobabilitiesforrealizationoftheeventondifferenttrialsaredifferenttheprobabilityPnmthattheeventoccursmtimesinntrialsequalsthecoefficientofumintheexpansionofthegeneratingfunction

whereqk=1ndashpkpkbeingtheprobabilitythattheeventoccursinthekthtrialThecoefficientsPnmcanbedeterminedbydifferentiatingthefunctionG(u)

whichgivesforexample


Example81Whatismoreprobableinplayingagainstanequaladversary(ifthegamecannotendina tie) towin(a) threegamesoutoffourorfiveoutofeight(b)atleastthreegamesoutoffouroratleastfiveoutofeight

SOLUTIONSincetheadversariesareequaltheprobabilitiesforthemtowinorloseagameareequaliep=q=

(a)Theprobabilityofwinningthreegamesoutoffouris

The probability of winning five games out of eight is732Since14gt732itismoreprobabletowinthree

gamesoutoffour(b)Theprobabilityofwinningatleastthreegamesoutoffouris

andtheprobabilityofwinningatleastfivegamesoutofeightis

Since93256gt516itismoreprobabletowinatleastfivegamesoutofeightSimilarlyonecansolveProblems81to831

Example 82 There are six consumers of electric current The probabilitythatundercertainconditionsabreakdownwilloccurthatwilldisconnectoneoftheconsumersis06forthefirstconsumer02forthesecondand03foreachoftheremainingfourFindtheprobabilitythatthegeneratorwillbecompletelydisconnected if (a) all the consumers are connected in series (b) all theconsumersareconnectedasshowninthescheme(Figure7)

FIGURE7

SOLUTION(a)Theprobabilitythatallsixconsumerswillnotbedisconnectedis equal to the product of the probabilities for each consumer not to bedisconnectedthatis

The required probability equals the probability that at least one consumerwillbedisconnectedthatisp=1ndashqasymp0923

(b) In thiscase thegenerator iscompletelydisconnected if ineachpairofsuccessivelyconnectedconsumersthereisatleastonewhoisdisconnected

Problems832to835canbesolvedsimilarly

Example83Alotcontains1percentofdefectiveitemsWhatshouldbethenumberof items ina randomsample so that theprobabilityof findingat leastonedefectiveiteminitisatleast095

SOLUTIONTherequirednumbernisgivenbytheformulangeln(1ndashP)ln(1ndashp)InthepresentcaseP=095andp=001Thusngeln005ln099asymp296


Example 84 A wholesaler furnishes products to 10 retail stores Each ofthemcansendanorderforthenextdaywiththeprobability04independentoftheorders from theother storesFind themostprobablenumberofordersperdayandtheprobabilityofthisnumberoforders

SOLUTIONHerewehaven=10p=04(n+1)p=44Themostprobablenumbermicroofordersequalstheintegralpartofthenumber(n+1)pthatismicro=4

Theprobabilityofgettingfourordersoutof10is


PROBLEMS

81Findtheprobabilitythatthelicensenumberofthefirstcarencounteredonagivendaywillnotcontain(a)a5(b)two5rsquos

All licensenumbershavefourdigitsrepetitionsofdigitsarepermittedandalldigitsappearwithequalprobabilityinallpositions

82Thereare10childreninafamilyIftheprobabilitiesofaboyoragirlareboth05findtheprobabilitythatthisfamilyhas(a)fiveboys(b)atleastthreebutatmosteightboys

83 From a table of random numbers one copies at random 200 two-digitnumbers(from00to99)Findtheprobabilitythatamongthemthenumber33

appears(a)threetimes(b)fourtimes84Consider thata libraryhasonlybooksinmathematicsandengineering

The probabilities that any reader will select a book in mathematics andengineering are respectively 07 and 03 Find the probability that fivesuccessivereaderswilltakebooksonlyinengineeringoronlyinmathematicsifeachofthemtakesonlyonebook

85TwolightbulbsareconnectedinseriesinacircuitFindtheprobabilitythat an increase in the voltage above its rated value will break the circuit ifundertheseassumptionstheprobabilitythatabulbburnsoutis04foreachofthetwobulbs

86TheeventBwill occur only if the eventA occurs at least three timesFindtheprobabilityforrealizationoftheeventB in(a)fiveindependenttrials(b) seven independent trials if the probability of occurrenceof the eventA inonetrialisequalto03

87AnelectricsystemcontainingtwostagesoftypeAonestageoftypeBandfourstagesoftypeCisconnectedasshowninFigure8FindtheprobabilityofabreakinthecircuitsuchthatitcannotbeeliminatedwiththeaidofthekeyKiftheelementsoftypeAareoutoforderwiththeprobability03theelementsoftypeBwiththeprobability04andtheelementsoftypeCwiththeprobability02

88 The probability that a unit must undergo repairs afterm accidents isgivenbytheformulaG(m)=1ndash(1ndash1ω)mwhereωistheaveragenumberofaccidentsbeforetheunitissubmittedforrepairsProvethattheprobabilitythatafterncyclestheunitwillneedrepairsisgivenbytheformulaWn=1ndash(1ndashpω)nwherepistheprobabilitythatanaccidentwilloccurduringonecycle

89Consider four independent trials inwhich the eventA occurswith theprobability 03 The eventB will occur with the probability 1 if the eventAoccursatleasttwiceitcannotoccuriftheeventAdoesnotoccuranditoccurswith a probability 06 if the eventA occurs once Find the probability of theoccurrenceoftheeventB

810 Consider 200 independent shots fired at a target under identicalconditionsandleadingto116hitsWhichvalue12or23fortheprobabilityofhitting in one shot is more probable if before the trial both hypotheses areequallyprobable

FIGURE8

811Evaluatethedependenceofatleastoneoccurrenceoftheeventsin10independenttrialsontheprobabilitypforrealizationoftheeventAineachtrialforthefollowingvaluesofp001005010203040506

812Theprobability thataneventoccursat leastonce in four independenttrialsisequalto059WhatistheprobabilityofoccurrenceoftheeventAinonetrialiftheprobabilitiesareequalinalltrials

813Theprobabilitythataneventoccursineachof18independenttrialsis02Findtheprobabilitythatthiseventwilloccuratleastthreetimes

814 The probability ofwinningwith one purchased lottery ticket is 002Evaluatetheprobabilitiesofwinningaprizewithnticketsforn=1102030405060708090100iftheticketsbelongtodifferentseriesforeachcase

815Giventhatalotteryticketwinsaprizeandthattheprobabilitiesthatthisprizeisabicycleorawashingmachinearerespectively003and002findtheprobability of winning at least one of these items with 10 winning ticketsselectedfromdifferentseries

816AgameconsistsofthrowingringsonapegAplayergetssixringsandthrows them until the first success Find the probability that at least one ringremainsunusediftheprobabilityofasuccessfulthrowis01

817 Find the probability of scoring at least 28 points in three shots firedfrom a pistol at a targetwith themaximal score of 10 points per shot if theprobabilityofscoring30pointsis0008Assumethatinoneshottheprobabilityofscoringeightpointsis015andlessthaneightpoints04

818TwobasketballplayerseachmaketwoattemptsatthrowingaballintothebasketTheprobabilitiesofmakingabasketateachthrowarerespectively06 and 07 Find the probability that (a) bothwill have the same numbers ofbaskets(b)thefirstbasketballplayerwillhavemorebasketsthanthesecond

819 The probability that a tubewill remain in good condition after 1000hoursofoperation is02What is theprobability that at leastoneoutof threetubeswillremainingoodconditionafter1000hoursofoperation

820Threetechniciansproduceitemsofexcellentandgoodqualitiesontheir

machines The first and second technicians make excellent items with theprobability 09 and the third technician with the probability 08 One of thetechnicianshasmanufactured eight itemsofwhich twoaregoodWhat is theprobabilitythatamongthenexteightitemsmadebythistechniciantherewillbetwogoodandsixexcellentitems

821Forvictory in avolleyball competition a teammustwin threegamesoutoffivetheteamsarenotequallymatchedFindtheprobabilitythatthefirstteamwillwineachgameifforequalchancesthisteammustgiveoddsof(a)twogames(b)onegame

822 A competition between two chess players is conducted under thefollowingconditionsdrawsdonotcountthewinneristheonewhofirstscoresfourpointsundertheassumptionthattheadversaryhasinthiscaseatmosttwopointsifbothplayershavethreepointseachtheonewhoscoresfivepointsfirstwins

Foreachoftheplayersfindtheprobabilityofwinningthecompetitioniftheprobabilitiesoflosingeachgameareintheratio32

823ApersonusestwomatchboxesforsmokingHereachesatrandomforoneboxortheotherAftersometimehefindsoutthatoneboxisemptyWhatistheprobabilitythattherewillbekmatchesleftinthesecondboxifinitiallyeachboxhadnmatches(Banachrsquosproblem)

824Theprobabilityofscoring10pointsis07andninepoints03Findtheprobabilityofscoringatleast29pointsinthreeshots

825Duringeachexperimentoneoftwobatterieswithpowersof120wattsand 200watts is connected in the circuit for one hour The probabilities of afavorable outcome of this experiment are 006 and 008 respectively Oneconsidersthattheresultofaseriesofexperimentshasbeenattainedifonegetsatleastonefavorableoutcomeintheexperimentwiththebatteryof200wattsoratleasttwofavorableoutcomeswiththebatteryof120wattsThetotalenergyconsumedinallexperimentscannotexceed1200wattsWhichbattery ismoreefficient

826AdevicestopsifthereareatleastfivedefectivetubesoftypeIandatleasttwodefectivetubesoftypeIIFindtheprobabilitythatthedevicewillstopiffivetubesaredefectiveandiftheprobabilitiesofadefectivetubeamongthetubesoftypeIandIIare07and03respectively

827 The probability of a dangerous overload of a device is 04 in eachexperimentFind theprobability that thisdevicewillstop in three independentexperimentsiftheprobabilitiesofastopinonetwoandthreeexperimentsare0205and08

828Theprobabilitythatanyofnidenticalunitstakespartinanexperiment

isp(plt1n)Ifagivenunitparticipatesintheexperimentsexactlyktimestheresult of these experiments is considered attained Find the probability ofattainingthedesiredresultinmexperiments

829Undertheassumptionsoftheprecedingproblemfindtheprobabilityofattaining the desired result in (2k ndash 1) experiments if the experiments arediscontinuedwhentheresulthasbeenattained

830 The probability that a device will stop in a trial is 02 How manydevicesshouldbetriedsothattheprobabilityofatleastthreestopsis09

831ApointAmustbeconnectedwith10telephonesubscribersatapointBEachsubscriberkeepsthelinebusy12minutesperhourThecallsfromanytwosubscribersareindependentWhatistheminimalnumberofchannelsnecessarysothatallthesubscriberswillbeservedatanyinstantwiththeprobability099

832 Four radio signals are emitted successively The probabilities ofreceptionforeachofthemareindependentofthereceptionoftheothersignalsandequalrespectively010203and04Findtheprobabilitythatk signalswillbereceivedwherek=01234

833Usingtheassumptionsoftheprecedingproblemfindtheprobabilityofestablishing a two-part radio communication system if the probability of thiseventisequalto02forthereceptionofonesignal06fortwosignalsand1forthreeandfoursignals

834Theprobabilitiesthatthreetubesburnoutarerespectively0102and03Theprobabilitiesthatadevicewillstopifonetwoorthreetubesburnoutare02506and09respectivelyFindtheprobabilitythatthedevicewillstop

835Ahunterfiresashotatanelkfromadistanceof100mandhitsitwiththeprobability05Ifhedoesnothit itonthefirstshothefiresasecondshotfromadistanceof150mIfhedoesnothittheelkinthiscasehefiresthethirdshotfromadistanceof200mIftheprobabilityofahitisinverselyproportionaltothesquareofthedistancefindtheprobabilityofhittingtheelk

836Howmanynumbersshouldbeselectedfromatableofrandomnumberstoensurethemaximalprobabilityofappearanceamongthemofthreenumbersendingwitha7

837Theprobabilityofscoring10hits inoneshot isp=002Howmanyindependent shots should be fired so that the probability of scoring 10 hits atleastonceisatleast09

838During one cycle an automaticmachinemakes 10 items Howmanycyclesarenecessarysothattheprobabilityofmakingatleastonedefectiveitemisatleast08iftheprobabilitythatapartisdefectiveis001

839Circlesofradius1cmhavetheircenterslocated60cmapartonalineSeveral linesof thiskindareplacedparallel toeachother inthesameplanea

relativeshiftofthelineswithanyamountfrom0to60cmisequallyprobableAcircleof radius7cmmoves in thesameplaneandperpendicularly to theselinesWhatshouldbethenumberoflinessothattheprobabilityofintersectionofthemovingcirclewithoneoftheothercirclesisatleast09

840Fromaboxcontaining20whiteandtwoblackballsnballsaredrawnwithreplacementoneata timeFind theminimalnumberofdrawsso that theprobabilityofgettingablackballatleastonceexceeds12

841Foracertainbasketballplayertheprobabilityofthrowingtheballintothe basket in one throw is 04 Hemakes 10 throws Find themost probablenumberofsuccessfulthrowsandthecorrespondingprobability

842Findthemostprobablenumberofnegativeandpositiveerrorsandthecorresponding probabilities in four measurements if in each of them theprobabilityofapositiveerrorequals23andofanegativeone13

9 THEMULTINOMIALDISTRIBUTIONRECURSIONFORMULASGENERATINGFUNCTIONS

BasicFormulasTheprobabilitythat inn independenttrials inwhichtheeventsA1A2hellip

Am occur with the corresponding probabilities p1 p2 hellip pm the events Ak

wherek=12hellipmwilloccurexactlyntimes isgivenbythemultinomialdistributionformula

TheprobabilityPnn1n2hellipnm isthecoefficientof in thefollowinggeneratingfunction

The generating function forn +N independent trials is the product of thegeneratingfunctionsfornandNtrialsrespectivelyUsingthispropertyonecanfrequently simplify the calculation of the required probabilities For the samepurpose one applies a proper substitution of the arguments in the generatingfunction If for instanceonewishes to find theprobability that inn trials the

event A1 will appear l times more than the event A2 then in the generatingfunction one should setu2 = 1u u1 = u uj = 1 where j = 3 4hellipm Therequiredprobabilityisthecoefficientofulintheexpansioninapowerseriesforthefunction

Ifpk=1mwherek=12hellipmandonewishestofindtheprobabilitythatthesumofthenumbersoftheoccurringeventsisronelooksforthecoefficientofurintheexpansioninpowersofuofthefunction

In the expansion ofG(u) it is convenient to use for (1 ndash u)ndashn the followingexpansion

Factorialsoflargenumberscanbeobtainedfromlogarithmtables(see2Tinthetablelist)orapproximatedbyStirlingrsquosformula

Theprobabilityofoccurrenceofagiveneventcansometimesbeobtainedusingrelations(recursionformulas)oftheform

whereakandbkaregivenconstantsTherequiredprobability isdeterminedbypassage from n to n + 1 after an evaluation based on initial data of theprobabilitiesforseveralvaluesofk


Example 91 The probabilities that the diameter of any item is less thangreater than or equal to some accepted value are respectively 005 010 and085From the total lot one selects100 randomsamplesFind theprobabilitythatamongthemtherewillbefiveitemswithasmallerdiameterandfivewithalargerdiameterthantheacceptablediameter

SOLUTIONLettheeventA1meanthatanitemofthefirsttypeanitemA2ofthesecondtypeandA3ofthethirdtypearerandomlyselectedByassumptionp1=005p2=010p3=085Thetotalnumberoftrialsnis100WeseektheprobabilitypthattheeventsA1andA2willoccurfivetimeseachThenn1=n2=5n3=90Thereforetherequiredprobability

Ifweuselogarithmswefind

Usingthelogarithmtableforfactorialsandthetablefordecimallogarithmsweobtain

SimilarlyonecansolveProblems91to97and925

Example92IneachtrialtheprobabilityofoccurrenceofaneventequalspWhatistheprobabilitythatthenumberofoccurrencesoftheeventwillbeeveninntrials

SOLUTIONLetusdenotebypk theprobability that ink trials theeventwilloccuranevennumberoftimes

Beforethekthtrialonecanmaketwohypothesesinthe(kndash1)st trial theevent occurred an even or odd number of times The probabilities of thesehypothesesarepkndash1and1ndashpkndash1respectivelyThen

thatis

Representingthelastexpressionintheform

and respectivelymultiplying the left and right sides ofn such equalitiesweobtain

Simplifyingbothsidesofthelastequalityby wefind

Sincep0=1therequiredprobabilitywillbe

Problems98to913and926canbesolvedsimilarly

Example 93 Find the probability of purchasing a ticket with a numberwhosesumsofthefirstthreeandlastthreedigitsareequalifithassixdigitsandmaybeanynumberfrom000000to999999

SOLUTIONLetusfirstconsiderthefirstthreedigitsofthenumberSincetheyare arbitrary one can consider thatoneperforms three trials (n=3) inwhichanyonedigitoccurswiththeprobabilityp=110

Inthegivencasethenumberofeventsmis10theprobabilityisp=110wherek=01hellip9andthegeneratingfunctionhastheform

wherethesubscriptkofukindicatesthenumberkoccurringinthetrial

Letussetuk=ukThenthecoefficientofuσintheexpansionofthefunction

gives theprobability that thesumof thefirst threedigitsof thenumberontheticketisσ

Similarlythecoefficientofundashσintheexpansionof

givestheprobabilitythatthesumofthelastthreedigitsofthenumberisσButinthiscasethecoefficientofu0intheexpansion

isequaltotherequiredprobabilitythatthesumofthefirstthreedigitsandthesumofthelastthreedigitsareequal

Wehave

Thustherequiredprobabilityis


PROBLEMS

91 Suppose that an urn contains three balls one black one red and onewhiteOnedrawsballsfromitfivetimesoneballatatimewithreplacementFind the probability that the red and white balls will be drawn at least twiceeach

92Anemployeeproducesagooditemwithprobability090anitemwithadefect that can be eliminated with the probability 009 and an item with apermanent defect with the probability 001 He makes three items Find theprobabilitythatamongthemthereisatleastonegooditemandatleastonewithadefectthatcanbeeliminated

93Eachofnineballscanbeplacedwithequalprobability inoneof threeinitially empty boxes Find the probability that (a) therewill be three balls ineachbox(b)therewillbefourballsinthefirstboxthreeinthesecondboxandtwointhethirdbox

94 Ten shots are fired at a target consisting of an inner circle and twoconcentricannuliTheprobabilitiesofhittingtheseregionsinoneshotare015022and013respectivelyFindtheprobabilitythattherewillbesixhitsinthecirclethreeinthefirstannulusandoneinthesecondannulus

95AdeviceconsistsoffourunitseachmadeofvacuumtubesIfonetubeisoutofordertheprobabilitiesthatitbelongstoagivenunitarep1=06111p2=p3 = 0664p4 = 02561 respectively and these do not depend on howmanytubeswerepreviouslyoutoforderFindtheprobabilitythatthedevicewillstopwhenfourtubesareoutoforderifthiseventmayoccurwhenatleastonetubeofthefirstunitoratleastonetubeineachofthesecondandthirdunitsisoutoforder

96TwelvepersonsgetonatrainthathassixcarseachpassengermayselectwithequalprobabilityeachofthecarsFindtheprobabilitythat(a)therewillbetwo passengers in each car (b) therewill be one carwithout passengers onewithonepassenger twowith twopassengerseachand theremaining twowiththreeandfourpassengersrespectively

97AnurncontainslwhitemblackandnredballsFromitaredrawnwithreplacementoneatatimel1+m1+n1ballsFindtheprobabilitythat(a)firstl1whiteballsthenm1blackballsandfinallynxredballsaredrawn(b)l1whitem1 black and n1 red balls are drawn so that balls of identical color appearsuccessivelybutthesuccessionofcolorsmaybearbitrary(c)l1whitem1blackandn1redballsaredrawninanysuccession

98 Find the probability that in n tosses a coin will show heads an oddnumberoftimes

99Twoequallymatchedadversariesplaychessuntiloneof themleadsbytwogamesWhatistheprobabilitythat2ndecisivegames(thatarenotdraws)willbeneeded

910TwopersonsplayuntiloneofthemwinsallthemoneyfromtheotherFindtheprobabilitythatexactlyngameswillbenecessaryifall thestakesareequal each player has at the beginning three stakes and the probability ofwinningagameis12foreachofthetwoplayers

911Twopersonsplayuntil oneof them is ruinedThe first playerhas aninitialcapitalofndollarsandthesecondmdollarsTheprobabilitiesofwinningarerespectivelypandq(p+q=1)Ineachgamethegainforoneplayer(lossfortheother)isonedollarFindtheprobabilitiesofacompleteruinforeachofthem

912Inachesscompetitiontherearen+1equallygoodplayersEachmanplayseachoftheothersuntilhelosesThecompetitioncontinuesuntiloneoftheplayerswins n gamesWhat is the probability thatm decisive games will beplayed(drawsarenotcounted)

913Acompetitionbetween twoequal chessplayers takesplaceunder thefollowingconditionsthedrawsarenottakenintoaccountthewinneristheonewho scores six points if his adversary scores nomore than four points if onewins six games and the other five then the competition continues until thedifferenceinpointsbecomestwo

Findtheprobabilitythatthenumberofdecisivegamesis(a)atmost10(b)exactlyn

914Theprobabilitythataneventoccursineachofnexperimentsisequaltop Prove that the generating function for the probabilities of at least n ndash moccurrencesofthiseventis

915Theprobabilitythataneventoccursinthecthexperimentisequaltopk(k = 1 2hellip n) Prove that the generating functions for the probabilities ofrespectivelyatmostmoccurrencesandatleastnndashmoccurrencesofthiseventinnindependenttrialsare

916Eachof twomarksmenfiresn shotsathis targetFind theprobabilitythattheywillscorethesamenumberofhitsiftheprobabilityofhittingineachshotis05

917Eachof twoidenticaldevices leftandrighthas twotubesAfter100hours of operation one tube can burn out in only one of them with theprobability14andbothtubescanburnoutwiththeprobability116Findtheprobabilitythatinnpairsofsuchdevicesthenumberofburnt-out tubesintheleftdeviceswillexceedatleastbym(mle2n)thenumberofburnt-outtubesintherightdevicesFindthisprobabilityinthecasewhenn=m=3

918 The competition for the title ofworld champion in 100 square-boardcheckersconsistsof20gamesFindtheprobabilitythatitwillendwiththescore128iftheprobabilityofwinningeachgameis02foreachofthetwoplayers

919 Towin the competition for the title ofworld champion in chess thechallengermustscoreatleast125pointsoutofapossible24Inthecaseofatie(1212) the title is kept by the defending champion The participants are twoequal players whose probabilities of winning a game are half as great as theprobabilitiesofa tieFind(a) theprobability that thedefendingchampionwillkeep his title and the probability that the challenger will become the worldchampion(b)theprobabilitythat20gameswillbeplayedinthiscompetition

920Findtheprobabilitythatinnthrowsofapairofdicethesumofpointsmarkedontheupperfaceswillbe(a)equaltoagivennumberm(b)notgreaterthanm

Findtheseprobabilitiesforn=10andm=20921Findtheprobabilityofgettingaticketwithanumberthesumofwhose

digitsis21ifallnumbersoftheticketfrom0to999999areequallyprobable922Any of the n quantitiesX1X2hellipXn can take any integral positive

valuefrom1tomwithequalprobabilityFindtheprobabilitythatthesumX1+X2+middotmiddotmiddot+Xnwillbe(a)equaltoagivennumberN(nmgeNgen)(b)notlessthanagivennumberN

923TwomarksmenfirethreeshotseachattheirtargetsOnecanscoreanynumberofpointsfromsevento10withequalprobabilitywhereasfortheothertheprobabilityof scoringsevenand10points is18 andof scoringeightandninepointsis38Findtheprobabilitythat(a)thefirstmarksmanwillscore25points (b) the secondmarksmanwill score 25points (c) bothmarksmenwillscorethesamenumberofpoints

924 Two distinguishable coins are tossed simultaneously and repeatedlyFind theprobability thatat thenth toss (andnotbefore)eachwillhaveshownheadsasmanytimesastheother

925FindtheprobabilitythatarunoffwillbenecessaryintheelectionsoflpersonsifnpeoplevoteTheprobabilityofbeingeliminatedisthesameforeachof thek candidates andequal top and to be elected a candidatemust get themajorityofthevotesArunofftakesplaceonlyinthecasewhencandidateslandl+1getanequalnumberofvotes

926Twoequalvolleyball teamsplayonegameThegamecontinuesuntiloneoftheteamsleadsbytwopointstheminimalscorenecessaryis15Findtheprobabilities(a)PkandQk that thegamewillbewonrespectivelyby thefirstteam(whichservestheballfirst)andthesecondteamwiththescore15k(k=0113)(b)P1andQ1thatthegamewillbewonbyeachoftheteamsifthelosingteamhasatmost13points(c)PkandQkthatthegamewillbewonwithascoreof(16+k)(14+k)wherek=01hellip(d)PIIandQIIthatthegamewillbewonifeachteamlosesatleast14points(e)PandQ thatthegamewillbewonrespectivelybythefirstandsecondteams

1Byaldquoramdomnumberrdquoherewemeanak-digitnumber(kgt1)suchthatanyofitsdigitsmayrangefrom0to9withequalprobability

2Therestrictionofequalprobabilityused informulatingseveralproblemswithapoint thathits theinteroirofnanypartofadomain(lineartwo-dimensionalandsoforth)isunderstoodonlyinconnectionwiththenotionofgeometricprobability

3Considerthatthenumeratoranddenomonatorarerandomlyselectednumbersfromthesequence12hellipkandsetkrarrinfin

II RANDOMVARIABLES

10 THEPROBABILITYDISTRIBUTIONSERIESTHEDISTRIBUTIONPOLYGONANDTHEDISTRIBUTIONFUNCTIONOFADISCRETERANDOMVARIABLE

BasicFormulasA random variable is said to be discrete if its possible values can be

enumeratedAdiscrete randomvariableX canbe specifiedby (1)adistribution series

(2)adistributionfunction(integraldistributionlaw)ByadistributionserieswemeanthesetofallpossiblevaluesxtofXandthe

corresponding probabilities pi = P(X = xi) A distribution series can berepresentedbyatable(seeTable2)oraformula

Theprobabilitiespisatisfythecondition

inwhichthevalueofnmaybefiniteorinfiniteThe graphic representation of a distribution series is called a distribution

polygonToconstructitonerepresentsthevaluesoftherandomvariable(xi)onthex-axis and the probabilitiespi on the y-axis next one joins the pointsAiwiththecoordinates(xipi)byabrokencurve(Figure9)

Thedistributionfunction(integraldistributionlaw)ofarandomvariableXisdefinedas thefunctionF(x)equal to theprobabilityP(Xltx) that therandomvariableislessthanthe(arbitrarilychosen)valuexThefunctionF(x) isgivenbytheformula

inwhichthesummationisextendedoverallvaluesofisuchthatxiltx

TABLE2

FIGURE9


Example101Fromalotof100itemsofwhich10aredefectivearandomsampleofsize5isselectedforqualitycontrolConstructthedistributionseriesoftherandomnumberXofdefectiveitemscontainedinthesample

SOLUTION Since the number of defective items in the sample can be anypositive integer from 0 to 5 inclusive the possible values xi of the randomvariableXare

TheprobabilityP(X=k)thatthesamplewillcontainexactlyk(k=012345)defectiveitemsis

The computations with the preceding formula give with an accuracy of0001thefollowingresults

Usingforverification theequality wecanconvinceourselvesthatthecomputationsandtheround-offarecorrect(seeTable3)

TABLE3

SimilarlyonecansolveProblems1013and1014

Example102 Items are tested under overload conditions The probabilitythat each item passes the test is 45 and independence prevails The tests areconcludedwhen an item fails tomeet the requirements of the testDerive theformulaforthedistributionseriesofthenumberoftrials

SOLUTIONThetrialsendwiththethitem(k=123hellip)if thefirstkndash1itemspassthetestandthekthitemfails

IfXistherandomnumberoftrialsthen

TheformulaobtainedforthedistributionseriesisequivalenttoTable4

TABLE4

The peculiarity of the current problem is that theoretically the number oftrialscanbeinfinitebuttheprobabilityofsuchaneventiszero

Problems 102 104 105 107 1010 and 1012 are solved in a similarmanner

Example103AcarhasfourtrafficlightsonitsrouteEachofthemallowsittomoveaheadorstopwiththeprobability05

Sketchthedistributionpolygonoftheprobabilitiesofthenumbersoflightspassedbythecarbeforethefirststophasoccurred

SOLUTION Let X denote the random number of lights passed by the carbeforethefirststopoccursitcanassumethefollowingvalues

Theprobabilitiespi=P(X=xi)thatthenumberoftrafficlightsXpassedbythecarwillequalsomegivenvaluecanbecomputedwiththeformula

inwhichp is theprobabilitywithwhich the traffic lightscanstop thecar(p=05)

Asa resultof thesecomputationsweobtain thatp1=05p2=025p3=0125p4=00625p5=00625Withtheseresultsweconstructtheprobabilitydistributionpolygon(Figure10)

FollowingthisexamplewecansolveProblems103108and109

Example104AspacerockethasadeviceconsistingoffourunitsA1A2A3

A4 eachofwhich fails tooperatewhenat leastoneelementaryparticlehits itThefailureoftheentiredeviceoccurseitherifA1failsorifA2A3andA4 failsimultaneously

FIGURE10

ConstructthedistributionfunctionF(x)oftherandomnumberofelementaryparticlesXforwhichtheentiredevicewillfailiftheprobabilitythataparticlereachingthedevicewillhitA1isp1=04andtheprobabilitiesforhittingA2A3andA4arerespectivelyp2=p3=p4=02

SOLUTIONLetA1A2A3A4denote theevents thatA1A2A3A4 failTherequired distribution function F(x) equals the probability that the device willcontinueitsoperationafternltxhitsie

Usingtheformula(seeSection5)

andapplyingtheformulafortheadditionofprobabilitiesweobtain

whereall theprobabilitiesaredefinedundertheassumptionthatnparticleshitthedeviceSincep1+p2+p3+p4=1andforeachhitofaparticleoneandonlyonestagenecessarilyfailstooperatewehave

Thustakingintoaccountthatp2=p3=p4=02weobtain

where[x]denotesthelargestintegerlessthanxforexample[59]=5[5]=4

FIGURE11

ThereforethegraphoftheprobabilitydistributionfunctionforseveralinitialvaluesofxhastheformshowninFigure11

Problems106and1011aresolvedsimilarly

PROBLEMS

101 Construct the distribution series and the distribution function for arandom number of successful events in one experiment if the experimentconsistsofthrowingaballintoabasketandtheprobabilityofasuccessinonetrialisp=03

102Anexperimentconsistsofthreeindependenttossingsofacoinineachofwhichheadsshowsupwiththeprobabilityp=05Forarandomnumberofheads construct (a) its distribution series (b) distribution polygon (c)distributionfunction

103FivedevicesaresubjectedtosuccessivereliabilitytestsEachdeviceistested only if the preceding one turns out to be reliable Construct thedistributionseriesofarandomnumberoftestsiftheprobabilityofpassingthesetestsis09foreachdevice

104 Some independent experiments are discontinued when the firstfavorableoutcomehasoccurredForarandomnumberofexperimentsfind(a)

thedistributionseries(b)thedistributionpolygon(c)themostprobablenumberofexperimentsiftheprobabilityofafavorableoutcomeineachtrialis05

105 Two basketball players shoot the ball alternately until one of themscoresConstructthedistributionseriesforarandomnumberofshotsthrownbyeachofthemiftheprobabilityofasuccessis04forthefirstplayerand06forthesecond

106Atargetconsistsofacirclenumbered1andtwoannulinumbered2and3Byhittingthecirclenumbered1onescores10pointstheannulusnumbered2 5 points and the annulus numbered 3 1 point The correspondingprobabilitiesofhittingthecirclenumbered1andannulinumbered2and3are0503and02Constructthedistributionseriesforarandomsumofscoresasaresultofthreehits

107Anexperiment isperformedwithaseriesof identicaldevices thatareturned on successively for a period of five seconds each The lifetime of onedevice is16secondsTheexperiment isdiscontinuedwhenat leastonedevicestops Find the distribution series for a random number of devices if theprobabilityofstoppingis12foreachdevice

108TherearenpatternsforthesameitemTheprobabilityofproducinganondefectiveitemfromeachofthemisp(a)Findthedistributionseriesof thenumberofpatterns leftafter thefirstnondefective itemhasbeenproduced (b)Constructthedistributionseriesforarandomnumberofpatternsused

109Alotofn items is tested for reliability theprobability thateach itempasses the test is p Construct the distribution series for a random number ofitemsthatpassthetest

1010Adeviceconsistingofunitsab1andb2failstooperateiftheeventC=AcupB1B2 whereA denotes the failure of the unit a andB1 and B2 denotefailureoftheunitsb1andb2respectivelyThefailuresoccurwhenthedeviceishitbyatleastonecosmicparticleConstructthedistributionseriesofanumberofrandomparticleshittingthedeviceiftheprobabilitiesthataparticlehitsoneoftheunitsareP(A)=05P(B1)=P(B2)=025

1011An experiment can be a successwith probabilityp or a failurewithprobability(1ndashp)TheprobabilityofafavorableoutcomeinmsuccessfultrialsisP(m)=1ndash(1ndash1ω)mConstructthedistributionseriesofthenumberoftrialsnecessaryforafavorableresult

1012 The number of trialsX is a random integer between 0 and infin TheprobabilityP(X=k)=(nkendashn)kEachtrialcanbeasuccesswiththeprobabilitypandafailurewith theprobability(1ndashp)Construct thedistributionseriesofthenumberofsuccessfultrials

1013Theprobabilityofobtainingheads ineachoffive tossesofacoin is05 Find the distribution series for the ratio of the numberX of heads to thenumberYoftails

1014 Construct the distribution series for the sum of digits of three-digitrandomnumbers

11 THEDISTRIBUTIONFUNCTIONANDTHEPROBABILITYDENSITYFUNCTIONOFACONTINUOUSRANDOMVARIABLE

BasicFormulasArandomvariable is said tobecontinuous if it canassumeanynumerical

valuesonagivenintervalandforwhichforanyxonthisintervalthereexiststhelimit

calledprobabilitydensityA continuous random variable can be defined either by a distribution

functionF(x)(theintegraldistributionlaw)orbyaprobabilitydensityfunctionf(x)(differentialdistributionlaw)

The distribution function F(x) = P(X lt x) where x is an arbitrary realnumbergivestheprobabilitythatarandomvariableXwillbelessthanx

ThedistributionfunctionF(x)hasthefollowingbasicproperties(1)P(aleXltb)=F(b)ndashF(a)(2)F(x1)leF(x2)ifx1ltx2

(3)

(4)

The probability density function (differential distribution law) f(x) has thefollowingfundamentalproperties

(1)f(x)ge0

(2)

(3)

(4)

ThequantityxpdefinedbytheequalityF(xp)=piscalledaquantilethequantilex05iscalledthemedianIfthedensityhasamaximumthevalueofxforwhichf(x)=maxiscalledthemode

Thenotionofprobabilitydensity f(x) can alsobe introduced for adiscreterandomvariablebysetting

in which xk denote the possible values of the random variable pk are theircorrespondingprobabilities

δ(x)istheδ-functionthatisaldquogeneralizedrdquofunctionwiththeproperties

whereφ(x)isanyfunctioncontinuousatthepointx=yThefunctionδ(x)canberepresentedanalyticallyby

wheretheintegralisunderstoodinthesenseofitsprincipalvalue1


Example111TheprojectionXoftheradius-vectorofarandompointonacircumference of radiusa onto the diameter has the distribution function (thearcsinelaw)

Determine(a)theprobabilitythatXwillbeontheinterval(ndasha2a2) (b)thequantilex075 (c) theprobabilitydensity f(x)of the randomvariableX (d)themodeandmedianofthedistribution

SOLUTION (a)TheprobabilitythatXassumesvalueson the interval (ndasha2a2)isequalto

(b)Byassumptionp=075solvingtheequation

weobtain

(c)Theprobabilitydensityf(x)oftherandomvariableXis(1)forallvaluesofxbelongingtotheinterval(ndashaa)

(2)zeroforalltheremainingvaluesofx

(d) We call the value of the argument for which the probability densityachievesitsmaximumthedistributionmodeThearcsinelawhasnomodesincethefunction

hasnomaximaWe call the quantity x05 the distribution median defined by the equality

F(x05)=12

Solvingtheequation

wefindthatx05=0Problems111to118aresolvedsimilarly

Example112Theprobabilitydensityofarandomvariableis

Find(a)thecoefficienta(b)thedistributionfunctionoftherandomvariableX(c)theprobabilitythattherandomvariablebelongstotheinterval(01k)

SOLUTION(a)Thecoefficientaisgivenbytheequality

Thisimpliesthat

Integratingbypartstwiceweobtain

Consequentlya=k32andtheprobabilitydensityhastheform

(b)ThedistributionfunctionF(x)oftherandomvariableXisdeterminedbytheformula

(c)TheprobabilityP(0ltX lt lk) that the randomvariableXwill assumevaluesonthegivenintervaliscomputedaccordingtotheformula


Example113Anelectronicdevicehasthreeparallel linesTheprobabilitythat each line fails to operate during thewarranty period of the device is 01Using the 8-function express the probability density for a random number oflines that fail tooperateduring thewarrantyperiod if thefailureofone line isindependentofwhethertheotherlinesoperate

SOLUTION Let us denote byX the random numbers of lines that fail TherandomvariableXisdiscreteanditsdistributionseries(Table5)is

TABLE5

Usingthenotionofprobabilitydensityforadiscretevariableweobtain

SimilarlywecansolveProblem1115

PROBLEMS

111ThedistributionfunctionofauniformlydistributedrandomvariableXhastheform

FindtheprobabilitydensityoftherandomvariableX112Giventhedistributionfunctionofarandomvariable

findtheprobabilitydensityoftherandomvariableX113Cramer(1946)givesthedistributionfunctionoftheyearlyincomesof

personswhomustpayincometax

Findtheyearlyincomethatcanbeexceededbyarandomlyselectedtaxpayerwiththeprobability05

114 The distribution function of the random period during which a radiodeviceoperateswithoutfailureshastheform

Find(a)theprobabilitythatthedevicewilloperatewithoutfailuresduringatimeperiodT(b)theprobabilitydensityf(t)

115 The random variable representing the eccentricity of an item ischaracterizedbytheRayleighdistribution

Find(a)themodeofthedistribution(b)themedianofthedistribution(c)theprobabilitydensityf(x)

116TheWeibulldistributionfunction

characterizes in a series of cases the lifetime of the elements of an electronicinstrument

Find (a) the probability density f(x) (b) the quantile of order p of thisdistribution(c)themodeofthedistribution

117The randomnonoperatingperiodofa radiodevicehas theprobabilitydensity

whereM=loge=04343hellip(thisisthelogarithmicnormaldistributionlaw)Find (a) themodeof thedistribution forx0=1and (b) the

distributionfunction118 Given the distribution function of a random variableXF(x) = a + b

arctan (x2) (ndash infin lt x lt + infin) (the Cauchy probability law) determine (a)constantsaandb(b)theprobabilitydensity(c)P(αleXltβ)

119Howlargeshouldabesothatf(x)=aendashx2istheprobabilitydensityofarandomvariableXvaryingbetweeninfinitebounds

1110Forwhichvalueofaisthefunction

equaltotheprobabilitydensityofarandomvariableXFind (a) the distribution function of the random variable X (b) the

probabilitythattherandomvariablewillfallintheinterval(ndash11)1111Thescaleofastopwatchhasdivisionsof02secondseachWhat is

theprobability that theerror in the timeestimate is larger than005seconds ifthe estimate ismadewith an accuracy of one divisionwith a round-off to thenearestinteger

1112Theazimuthal limbhasdivisionsof1degeachWhat is theprobabilitythat therewill occur an error ofplusmn10prime in the computationof the azimuth if theangleestimatesareroundedofftothenearestdegree

1113ItisknownthattheprobabilityoffailureforanelectronictubeduringΔx days is kΔxwith a precision of higher order ofmagnitude thanΔx and isindependentofthenumberxofdaysduringwhichthetubeoperatespriortotheintervalΔxWhatistheprobabilityoffailureforatubeduringldays

1114AstreetcarlinehasalengthLTheprobabilitythatapassengerwillgetonthestreetcar inthevicinityofapointx isproportionaltox(Lndashx)2and theprobability that a passenger who entered at point x will get off at point y isproportionalto(yndashx)hhge0

Find the probability that (a) the passengerwill get on the streetcar beforepointz (b) thepassengerwhogoton the streetcar atpointxwill get off afterpointz

1115Somedevices are subjected to successive accelerated reliability teststhatareterminatedwhenthefirstfailureoccursUsingtheconceptofprobabilitydensityofadiscrete randomvariable find theprobabilitydensityofa randomnumberofdevicestestediftheprobabilityoffailureforeachdeviceis05

12 NUMERICALCHARACTERISTICSOFDISCRETERANDOMVARIABLES

BasicFormulas

Themostfrequentlyusedcharacteristicsofdiscreterandomvariablesarethemomentsofthesevariables

Themomentsmk and the central moments μk of the kth order of discreterandomvariablesaredefinedbytheformulas

inwhichM[Xk]istheexpectationofXkxiarethepossiblevaluesofarandomvariableXpi the probabilities of these values and is the expectation ofXThereforethefirstmomentisdeterminedbytheformula

thesecondcentralmomentorthevarianceisgivenby

orby

Themean-squaredeviationaisgivenbytherelation

IftheprobabilitiesofdifferentvaluesofXdependonthedisjointormutuallyexclusiveeventsAkthentheconditionalexpectationofXwiththeconditionthatAkoccursis

IfAk(k=12hellipm)formacompletesetofeventsthatis thenthetotalexpectationofXandtheconditionalexpectationarerelatedbytheformula

Inalltheprecedingformulasthenumberoftermsinthesumscanbeinfinitein this case for the existence of the expectation the sum must convergeabsolutely


Example121Fromalotcontaining100itemsofwhich10aredefectiveasample of five items is selected at random for quality control Find theexpectationforthenumberofdefectiveitemscontainedintherandomsample

SOLUTION The randomnumber of defective items contained in the samplehasthefollowingpossiblevalues

Theprobabilitypi=P(X=xi)thatXwillassumeagivenvaluexiis(seeExample101)

Therequiredexpectationis

Since isthecoefficientofu5intheproduct(1+u)10(1+u)90wesee isthecoefficientofu5intheexpression

Consequentlywehave


Example 122 A discrete random variable X is given by the distributionseriespk=P(X=k)k=123hellipExpresstheexpectationofXintermsofthegeneratingfunctionG(u)(seeSection9)

SOLUTIONBythedefinitionoftheexpectationofarandomvariable

On the other hand the value of the derivative of the generating functioncomputedatu=1is

Consequently

OnecansolveProblems123to126and1224to1226similarly

Example123Anexperimentcanbeasuccesswiththeprobabilitypandafailurewiththeprobability1ndashp

The conditional probability P(m) for achieving the desired result aftermsuccessfultrialsis

Find the expectation of the number of independent trials necessary forachievingthedesiredresult

SOLUTIONLetPn(A)denotetheprobabilityofachievingthedesiredresultinntrialsIfPnmistheprobabilityofexactlymsuccessesoutofatotalofntrialsthenaccordingtotheformulaforthetotalprobabilitywehave

Sincethetrialsareindependentandtheprobabilityofasuccessfuloutcomeineachofthemisp

SubstitutingintotheformulaforPn(A)thevaluesofPnmandp(m)weobtain

Toattainthedesiredresultexactlyntrialsarenecessaryifitwillbeattainedat thenth trialTheprobabilityof the latter circumstance isPn(A)ndashPn ndash 1(A)ConsequentlyM[X]theexpectationoftherandomnumberoftrialsnecessarytoattainthedesiredresultis

Tocomputethelastsumwemakeuseoftheequality

validfor|x|lt1Heresettingx=1ndashpωweobtain

SimilarlyProblems1210to12151221and1231canbesolved

Example124AdevicehasnfusesInthecaseofoverloadoneofthefusesburns out and is replaced by a newoneWhat is the expectationM[N] of thenumberofoverloadsNafterwhichalltheinitialfusesofthedevicearereplacedbynewonesifoneassumesthatitisequallylikelyforallfuses(oldornew)toburnout

SOLUTION Let us denote by M[N|k] the expectation of the number ofoverloadsafterwhichalltheinitialfuseswillbereplacedifkfuseshavenotyetbeenreplaced

TocomputeM[N|K]weusetheformulaforthetotalexpectationIfk fuses(k ge 1) remain nonreplaced then in order that one of them burns out asubsequent overload is necessary The average number of overloads necessaryfor a remaining fuse to burn out will depend on the result of the subsequentoverload

InthesubsequentoverloadtherecanoccurtwoeventsA1thatoneoftheinitialfusesburnsoutwiththeprobabilityP(A)1=knA2thatoneofthereplacedfusesburnsoutwiththeprobabilityP(A2)=1ndash

knIfatthesubsequentoverloadA1occurs thentheexpectationofthenumber

of overloads necessary for the replacement of all k fuses that have not beenreplacedbeforethisoverloadis1+M[N|kndash1]IfatthesubsequentoverloadA2occurs thentheexpectationequals1+M[N|k]Usingtheformulaforthetotalexpectationwefind

oraftersimpletransformations

Ifk=1 that isonlyone fusehasnotbeen replaced theprobabilityof itsreplacementequals1nThereforeaccordingtoExample123weshallhave

Thuswehaveachainofequalities

whosesumgives

or

Problems121612201222and1223canbesolvedinasimilarmanner

Example 125 As a result of experiments with two devicesA andB onefinds the probability of observing a noisewhose level is evaluated in a three-pointsystem(seeTable6)

TABLE6

Using the data fromTable6 select the better device ie the devicewithlowernoiselevel

SOLUTIONLetXdenotetherandomnoiselevelTheaveragenoiselevelforthedeviceAis

ForthedeviceB

Thuscomparedaccordingtotheaveragenumberofpointsbothdevicesareequivalent

Asanadditionalcriterionforcomparisonweusethemean-squaredeviationofthenoiselevel

Hence A gives a more stable indication with respect to the means andconsequentlyitisbetterthanB

PROBLEMS

121Findtheexpectationofthenumberofdevicesfailinginreliabilitytestsifineachtestonlyonedeviceistestedandtheprobabilityofitsfailureisp

122Assumingthatthemassofabodycantakewithequalprobabilityanyintegral number of grams on the interval 1 to 10 determine forwhich of thethreesetsofweights (a)122510 (b)123410 (c)112510 theaveragenumberofnecessaryweightswillbeminimumifonecanplaceweightsonlyononescaleandtheselectionofweightsismadetominimizethenumberusedintheprocessofweighing

123Acertaindevice consistingof five elements is testedTheprobabilitythatanelementnumberedifailstooperateis

Findtheexpectationandthevarianceofthenumberofelementsthatstopifthefailuresoftheelementsareindependent

124ThreedevicesaretestedindependentlyTheprobabilitiesoffailureforeach device are p1p2 and p3 respectively Prove that the expectation of thenumberofdevicesfailingtooperateisp1+p2+p3

125Determine theexpectationof thenumberofdevices failing tooperateduring a test period if the probability of failure for all devices is p and thenumberofdevicesthataretestedisn

126Alotterydistributesm1prizesworthklm2k2hellipmnandknThetotalnumberofticketsisNWhatshouldbethecostofaticketthattheexpectationofaprizeperticketisequaltohalfitscost

127The firstplayer tosses three fair coinsand the second two fair coinsThewinnerwhogets all fivecoins is theonewho scoresmoreheads In thecase of a tie the game is repeated until there is a decisive resultWhat is theexpectationofwinningforeachoftheplayers

128ThreepersonsAB andC play a game as follows two participate ineachgamethelosercedeshisplacetothethirdpersonthefirstgameisplayedbyAandBTheprobabilityofwinningeachgameis12foreachplayerTheycontinue to play until one of themwins two games in succession and getsmdollarsWhatistheexpectationofagainforeachoftheplayers(a)afterthefirst

gameundertheassumptionthatAwonit(b)atthebeginningofthegame129 Three persons A B and C play a game as follows two players

participateineachgamethewinnercedeshisplacetothethirdpersonfirstAplayswithBTheprobabilityofwinningeachgameis12foreachplayerTheycontinuetoplayuntiloneofthemwinstwoconsecutivetimesandgetsasumofmoney equal to the number of all games played What is the expectation ofwinningforAandCatthebeginningofthegame

1210 An automatic line in a state of normal adjustment can produce adefective item with probability p The readjustment of the line is madeimmediately after the first defective itemhasbeenproducedFind the averagenumberofitemsproducedbetweentworeadjustmentsoftheline

1211 The probability that a call signal emitted by one radio station isreceivedbyanother is02ateachemissionThecall signalsareemittedeveryfive secondsuntil an answer signal is receivedThe total passage time for thecallandanswersignals is16secondsFind theaveragenumberofcall signalsemittedbeforeatwo-wayconnectionhasbeenestablished

1212Findtheexpectationandthevarianceofthenumberofitemsproducedbetween two readjustments in aproduction line innormal adjustment if in thestate of normal adjustment the probability of a defective item is p and thereadjustmentismadeafterthekthdefectiveitemhasbeenproduced

1213 The conditional probability that a device stops computed under theassumptionthatmelementsfailtooperatehastheform

(a)forthedeviceA

(b)forthedeviceB

FIGURE12

Findtheexpectationofthenumberofnonoperatingelementsthat leadtostopsofthedevicesAandB

1214AblockingschemeconsistingoftherelayAconnectedinserieswithtworelaysBandCwhichareconnectedinparallelmustensuretheclosingofthecircuitbetweentheterminalsIandII(Figure12)AsaresultofdamagetherelayA can stopwith the probability 018 and the relaysB andCwith equalprobabilities022Findtheaveragenumberoftimesthattheschemeisturnedonuntilthefirstfailureoccurs

1215 A certain device contains the elements A B andC which can beaffectedbycosmicradiationandstopoperatingifatleastoneparticlehitsthemThe stoppageof thedeviceoccurs in thecaseof failureof theelementA or asimultaneousfailureoftheelementsBandCFindtheexpectationofthenumberofparticlesthatcausedthestoppageofthedeviceiftheconditionalprobabilitiesthataparticlereachingthedevicehitstheelementsABandCare0102and02respectively

1216AcertaindevicehasnelementsoftypeAandmelementsoftypeBIfone element of type A ceases to operate it is not replaced and the devicecontinuestooperateuntilthereremainsatleastonenondefectiveelementoftypeATheelementsoftypeBarereplacedrepeatedlyiftheyfailsothatthenumberofnondefectiveelementsoftypeBremainsconstantintheschemeThefailuresof each of the nondefective elements of the device are equally probableDeterminetheaveragenumberofelementfailuresleadingtoatotalstoppageofthedeviceietononoperationofallthenelementsoftypeA

1217Provethatthevarianceofthenumberofoccurrencesofaneventinthecaseofasingleexperimentdoesnotexceed14

1218 Find the conditions under which the third central moment of thebinomialdistributioniszero

1219 The distribution function of a random variable X is given by theequality

Provethatiflimnrarrinfinnp=athenlimnrarrinfinD[X]=a1220Tenballsaredrawninsuccessionfromanurncontainingaverylarge

number of white and black balls mixed in equal proportion The balls drawnbeforethefirstblackballoccursarereturnedtotheurnthefirstblackballthatappears togetherwith all those that follow is placed in another urnwhich is

initiallyemptyFind theexpectationof thenumberofblackandwhiteballs inthesecondurn

SolvethesameproblemundertheassertionthatthenumbernofballsdrawnisrandomandobeysPoissonrsquoslawwithparametera=10thatis

1221AgameconsistsoftossingafaircoinuntilheadsshowsupIfheadsappears at the kth tossing playerA gets k dollars from playerB HowmanydollarsshouldApaytoBbeforethegamestartssothattheexpectationoflossforeachplayeriszero(iethegameisldquofairrdquo)

1222AmotortransportcolumncanarriveataservicestationatanyinstantoftimeIfnrepairmenarescheduledondutybymethodAtheaveragenumberofcarsservicedequalsnpIftheyarescheduledbymethodBthenumbern[1ndash(1ndashp)2]willbeservicedifthecolumnarrivesduringthefirsttwoquartersof24hoursnpifthecolumnarrivesduringthethirdquarterof24hoursand05npifthecolumnarrivesduringthelastquarterof24hours

ForwhatvaluesofpshouldoneprefertheschedulingbymethodB1223A repairman servicesn one-typemachineswhich are in a row at a

distanceaapartfromoneanotherAfterfinishingtherepairononemachinehemovesontothemachinethatneedsservicebeforealltheothersAssumingthatmalfunctionsofallmachinesareequallyprobablecomputetheaveragedistancethisrepairmanmoves

1224 A random variable X may assume positive integral values withprobabilitiesdecreasinginageometricprogressionSelectthefirsttermandtheratio of the progression so that the expectation of X is 10 and under thisassumptioncomputetheprobabilityP10thatXle10

1225ArandomvariableXcanassumeanyintegralpositivevaluenwithaprobabilityproportionalto13nFindtheexpectationofX

1226Anexperiment isorganizedso thatarandomvariableXassumes thevaluelnwiththeprobability1nwherenisanypositiveintegerFindM[X]

1227Agameconsistsof repeated independent trials inwhich theeventAcanoccurwiththeprobabilitypIfAoccursinngt0consecutivetrialsanddoesnot occur at the (n + l)st trial the first player getsyn dollars from the secondplayer Ifn = 0 the first player pays one dollar to the secondDetermine thequantityyundertheassumptionthatthegamewillbeldquofairrdquoietheexpectationofagainforbothplayersis0Considerthecasewhenp=113

1228BallsaredrawnfromaboxcontainingmwhiteandnblackballsuntilawhiteballappearsFindtheexpectationofthenumberofballsdrawnanditsvarianceifeachballisreturnedtotheboxaftereachdraw

1229Consider twoboxeswithwhite andblackballs the first containsMwhiteballsoutofatotalofNandthesecondcontainsM1whiteballsoutofatotalofN1ballsAnexperimentconsistsofasimultaneousrandomdrawingofoneball fromeachboxand transfer to theotherboxafterwhich theballsaremixedDetermine theexpectationof thenumberofwhiteballs in thefirstboxafteragivennumberofktrialsConsiderthecasewhenkrarrinfin

1230 Communication with a floating research station is maintained by nradio stations The station that enters in a two-way connection is the one thatfirstreceivesthecallsignalsfromthefloatingstationandtheoccurrenceofthisevent is equallyprobable for eachof the radio stations (p = 1n)The floatingresearchstationwillcommunicatemtimesDeterminetheprobabilitythatradiostationNo1willbeinvolvedktimesFindtheexpectationandthevarianceofthenumberoftimesradiostationNo1communicates

1231TheindependenttrialsofadevicearerepeateduntilastopoccursTheprobabilityp of a stop is the same for each trialFind the expectationand thevarianceofthenumberoftrialsbeforestop

1232 Two persons toss a coin in turn until both get the same number ofheadsTheprobabilitythatafter2n tossingsbothwillhaveanequalnumberofheadsis

Determinetheexpectationofthenumberoftosses

13 NUMERICALCHARACTERISTICSOFCONTINUOUSRANDOMVARIABLES

BasicFormulasThe expectation =M[X] and the varianceD[X] of a random variableX

withtheprobabilitydensityf(x)canbecomputedbytheformulas

InthefirstcaseitisassumedthattheintegralconvergesabsolutelyThe expectation and thevarianceof continuous randomvariables have the

sameproperties as the analogousquantities fordiscrete randomvariablesThemean-squareorstandarddeviationσisdefinedbytheformula

For a symmetric distribution law one may define as a dispersioncharacteristic of a random variable the mean deviation E determined by thecondition

Themomentofkthordermkandthecentralmomentofkthorderμkcanbecomputedaccordingtotheformulas


Example131Theprobabilitydensityfortherandomrollingamplitudesofashiphastheform(Rayleighrsquoslaw)

Determine (a) the expectationM[X] (b) thevarianceD[X] and themean-

squaredeviationσ(c)thecentralmomentsofthirdandfourthorderμ3andμ4

SOLUTION The computation of the moments reduces to the evaluation ofintegralsoftheform

whichforevennare

where

andforoddn

(a)Theexpectationofarandomrollingamplitudeis

Performingthesubstitution weobtain

Thus

(b)Since

then

(c)

whereConsequently

wherem4=8a4J5=8a4Hence


Example 132 Find the mean deviation of a random variable whoseprobabilitydensity(theLaplacedensity)hastheform

SOLUTIONSincetheprobabilitydensityissymmetricwithrespecttozeroitfollowsthat =0ThemeandeviationEiscomputedaccordingtotheformula

FromthisitfollowsthatE=In2=06931InasimilarwayProblems131and134canbesolved

PROBLEMS

131TheprobabilitydensityofarandomvariableXhastheform

Determine (a)M[X] and (b)D[X] (c) find the relation between themean-squareandmeandeviationsofX

132ThedistributionfunctionofarandomvariableXhastheform

FindtheconstantsaandbComputeM[X]andD[X]133Determine theexpectationand thevarianceofa randomvariableX if

theprobabilitydensityis

134TheprobabilitydensityofarandomvariableXhastheform(thearcsinelaw)

Determinethevarianceandthemeandeviation135Theprobability density of the random rolling amplitudes of a ship is

givenbytheformula(Rayleighrsquoslaw)

inwhichσ2isthevarianceoftheangleofheelAretheamplitudessmallerandgreaterthantheaverageencounteredwiththe

samefrequency136 The velocities of themolecules of a gas have the probability density

(Maxwellrsquoslaw)

Find the expectation and thevarianceof thevelocityof themolecules andalsothemagnitudeofAforgivenh

137TheprobabilitydensityofarandomvariableXisgivenintheform

FindM[X]andD[X]138 Find the expectation and the variance of a random variable whose

probabilitydensityhastheform


probabilitydensityhastheform(theLaplacedensity)

1310ArandomvariableXhastheprobabilitydensity(thebeta-density)

DeterminetheparameterA theexpectationandthevarianceoftherandomvariableX

1311ArandomvariableXhastheprobabilitydensity(beta-density)

Find the parameter A the expectation and the variance of the randomvariableX

1312ArandomvariableXhastheprobabilitydensity

wherengt1isapositiveintegerDeterminetheconstantAtheexpectationandthevarianceoftherandomvariableX

1313The probability density of a nonnegative randomvariableX has theform

inwhichngt1FindAtheexpectationandthevarianceofX1314Provethatiftheconditions

aresatisfiedthenfortheexpectationofarandomvariablethefollowingequalityholdstrue

1315Theprobabilityoffindingasunkenshipduringasearchtimetisgivenbytheformula

Determinetheaveragetimeofsearchnecessarytofindtheship1316Findtheexpectationm(t)ofamassofradioactivesubstanceaftertime

t if initially themass of the substancewasm0 and the probability of nucleardisintegrationofanyatomperunittimeisaconstantp

1317 Find the half-life of a radioactive substance if the probability ofnuclear disintegrationof any atomperunit time is a constantp (Thehalf-lifeperiodTnisdefinedastheinstantwhenthemassoftheradioactivesubstanceisone-halfitsinitialvalue)

1318Theprocessingoftheresultsobtainedinacensushasshownthatthedifferentialdistribution lawof theagesofpersons involved in researchcanberepresentedbytheformula

Determine how many times the number of scientific workers under theaverageageexceedsthoseabovetheaverage

1319DetermineforStudentrsquosdistributiongivenbytheprobabilitydensity

themomentsmkforkltn1320ArandomvariableXobeysthebeta-densityieithastheprobability

density

Findthemomentofkthorder1321Findtheexpectationandthevarianceofarandomvariablehavingthe

probabilitydensity2πcos2xontheinterval(ndashπ2π2)1322Expressthecentralmomentμkintermsofthemoments1323 Express the moment mk in terms of the central moments and the

expectation

14 POISSONrsquoSLAW

BasicFormulasThedistributionseriesofarandomvariableXhastheform

inwhicha=M[X]iscalledthePoissondistributionlawPoissonrsquoslawcanapproximatelyreplacethebinomialdistributioninthecase

whentheprobabilitypofoccurrenceofaneventAineachtrialissmallandthenumbernoftrialsislargeInsuchacasetheapproximateequality

inwhicha=npholdstrue


Example 141 A radio device consists of 1000 electronic elements Theprobability of nonoperation for one element during one year of operation is0001 and is independent of the condition of the other elementsWhat is theprobabilitythatatleasttwoelementswillfailtooperateduringayear

SOLUTION Assuming that the random numberX of nonoperating elementsobeysPoissonrsquoslaw

wherea=np=1000middot0001=1weobtainthefollowing(1)theprobabilitythatexactlytwoelementsfailtooperateis

(2)theprobabilitythatatleasttwoelementsfailtooperateis


Example142Anexplosionofaballoonduringa reliability testgenerates100fragmentsthatareuniformlydistributedinaconeboundedbyanglesof30degand 60deg (Figure 13) Find the expectation and the variance of the number offragmentsreaching1sqmofthesurfaceofthespherelocatedinsidetheconeifthe radius of the sphere is 50 m and its center coincides with the point ofexplosion

SOLUTION Let a sphere of radius 50 m intersect the cone formed byfragments and let us determine the expectation of the number of fragmentspassing throughaunitareaof thesphericalzoneformedby the intersectionoftheconewiththesphereLetSdenotetheareaofthiszone

Since the total number of fragments is N = 100 the expectation for afragmentspassingthroughaunitareaofthesurfaceofthesphericalzonewillbe

TheprobabilitythatagivenfragmentwillreachagivenareaS0=1sqmissmall(itequalsS0S=175middot10ndash4) thereforeonemayconsider that therandomnumberoffragmentsreaching1sqmofthesurfaceofthesphereisdistributedaccordingtoPoissonrsquoslawandconsequentlythefollowingequalityisvalid

FIGURE13

InasimilarwayonecansolveProblems1410and1412

PROBLEMS

141Theexpectationforthenumberoffailuresofaradiodeviceduring10000hoursofoperationis10Findtheprobabilitythatthedevicefailstooperateduring100hours

142 The probability that any telephone subscriber calls the switchboardduringonehouris001Thetelephonestationservices300subscribersWhatistheprobabilitythatfoursubscriberswillcalltheswitchboardduringonehour

143Adevicecontains2000equallyreliableelementswiththeprobabilityoffailure for each of them equal top = 00005What is the probability that thedevicewill fail to operate if failure occurswhen at least one element fails tooperate

144Aswitchboardreceivesanaverageof60callsduringonehourWhatistheprobabilitythatduring30secondsinwhichtheoperatorisawaytherewillbenocalls

145Theprobability that an itemwill fail to pass a test is 0001Find theprobabilitythatfromatotalof5000itemsmorethanoneitemwillfailComparethe results obtained using Poissonrsquos distribution with those obtained with thebinomial distribution In the latter make use of logarithm tables with sevensignificantdigits

146Duringacertainperiodof time theaveragenumberofconnections towrongcallspertelephonesubscriberiseightWhatistheprobabilitythatforapreassigned subscriber the number of wrong connections will be greater thanfour

147Findtheprobabilitythatamong200itemstestedmorethanthreewillturnouttobedefectiveiftheaveragepercentageofdefectiveitemsis1percent

148 The proofs of a 500-page book contain 500 misprints Find theprobabilitythatthereareatleastthreemisprintsperpage

149 In the observations made by Rutherford and Geiger a radioactivesubstance emitted an average of 387 α-particles during 75 seconds Find theprobabilitythatthesubstancewillemitatleastoneα-particlepersecond

1410DeterminetheasymmetrycoefficientofarandomvariabledistributedaccordingtoPoissonrsquoslaw(TheasymmetrycoefficientisthequotientSk=μ3σ3)

1411Duringitsflightperiodtheinstrumentcompartmentofaspaceshipisreachedbyrelementaryparticleswiththeprobability

Theconditionalprobability foreachparticle tohitapreassignedunitequalspFindtheprobabilitythatthisunitwillbehitby(a)exactlykparticles(b)atleastoneparticle

1412Findthevarianceforthenumberofatoms(ofaradioactivesubstance)thatdecayinaunittimeifthemassofthesubstanceisMthehalf-lifeisTptheatomicweightisAandthenumberofatomsinagram-atomicweightisN02

1413DeterminetheprobabilitythatascreenofareaS=012sqcmlocatedat a distance r = 5 cm perpendicular to the flow of α-particles emitted by aradioactivesubstanceishitduringonesecondby(a)exactly10α-particles (b)not less than two α-particles if the half-life of the substance is Tn = 44middot109

yearsthemassofthesubstanceism=01gandtheatomicweightisA=23821414Provethatthemultinomialdistribution

inwhich

and

canbeapproximatedbythemultidimensionalPoissonlaw

inwhichλi=npi ifall theprobabilitiespiexcept forpm + 1 are small andn islarge

15 THENORMALDISTRIBUTIONLAW

BasicFormulasThe probability density of a normally distributed random variable has the

form

or

inwhichσ is themean-squaredeviation is themeandeviation(sometimesalsocalledldquoprobabledeviationrdquo)andρ=0476936hellip

The probability that a normally distributed random variable X assumesvalues on the interval (x1x2) can be computedbyusingoneof the followingformulas

inwhich

istheLaplacefunction(probabilityintegral)

inwhich

isthenormalizedLaplacefunction

ThevaluesofthefunctionsΦ(x)and aregivenin8Tand11Tinthetablelistonpages471472


Example 151 The measurement of the distance to a certain object isaccompaniedby systematic and randomerrorsThe systematic error equals50m in thedirectionofdecreasingdistanceThe randomerrorsobey thenormaldistribution law with the mean-square deviation σ = 100 m Find (1) theprobability of measuring the distance with an error not exceeding 150 m inabsolutevalue(2)theprobabilitythatthemeasureddistancedoesnotexceedtheactualone

SOLUTIONLetX denote the total errormade inmeasuring thedistance Itssystematiccomponentis =ndash50mConsequentlytheprobabilitydensityofthetotalerrorshastheform

(1)Accordingtothegeneralformulawehave

Theprobabilityintegralisanoddfunctionandhence

Fromthisweget

From8Tinthetablelistwefind

andfinally

(2)Theprobabilitythatthemeasureddistancewillnotexceedtheactualoneis

SinceΦ(infin)=limxrarrinfinΦ(x)=1andfrom8Tinthetablelistpage471wefindΦ(05)=03829itfollowsthat

SimilarlyonecansolveProblems151to154and1510to1514

Example152Determinethemeanerrorofaninstrumentwithnosystematicerrorsandwhoserandomerrorsaredistributedaccordingtothenormallawandfallwiththeprobability08withintheboundsplusmn20m

SOLUTIONFromtheassumptionoftheproblemitfollowsthat

Sincetheprobabilitydensityoftherandomerrorsisnormaland =0(thesystematicerrorisabsent)wehave

Theunknownvalue of themean error is determined as the solutionof thetranscendentalequation

Using11Tinthetablelistonpage472wefind

fromwhichitfollowsthat


PROBLEMS

151Ameasuring instrument gives a systematic error of 5m and ameanerrorof50mWhat is theprobability that theerrorofameasurementwillnotexceed5minabsolutevalue

152Thesystematicerrorinmaintainingthealtitudeofanairplaneis+20mandtherandomerrorischaracterizedbyameandeviationof50mForaflighttheplaneisassignedacorridor100mhighWhataretheprobabilitiesthattheplane will fly below inside and above the corridor if the plane is given analtitudecorrespondingtothemidpointofthecorridor

153Themeanerrorindistancemeasurementswitharadardeviceis25mDetermine(a)thevarianceoftheerrorsofthemeasurements(b)theprobabilityofobtainingerrorsnotexceeding20minabsolutevalue

154Ameasuring instrumenthasameanerrorof40mandnosystematicerrorsHowmanymeasurementsshouldbeperformedsothatinatleastoneofthemtheerrorwillnotexceed75minabsolutevaluewithaprobabilitygreaterthan09

155Given two randomvariablesX andY with equal variance one beingdistributednormallyandtheotheruniformlyfindthecorrelationbetweentheirmeandeviations

156AnormallydistributedrandomvariableXhastheexpectation =ndash15mandthemeandeviation10mComputethetableforthedistributionfunctionforvaluesoftheargumentincreasingby10mandplotthegraph

157AnaltimetergivesrandomandsystematicerrorsThesystematicerroris+20mandtherandomerrorsobeythenormaldistributionlawWhatshouldbethemeanerroroftheinstrumentsothattheerrorinaltitudemeasurementislessthan100mwiththeprobability09

158Findtherelationbetweenthearithmeticmeandeviation

ofanormallydistributedrandomvariableanditsmean-squaredeviation159For anormallydistributed randomvariableXwithM[X]=0 find (a)

P(Xgekσ)(b)P(|X|gekσ)(fork=123)1510Thegunpowderchargeofashotgunisweighedonscaleswithamean

errorof100mgThenominalmassofthegunpowderchargeis23gDeterminethe probability of damaging the gun if the maximum admitted mass of thegunpowderchargeis25g

1511Twoindependentmeasurementsaremadewithaninstrumenthavingameanerrorof20mandasystematicerrorof+10mWhatistheprobabilitythatbotherrorswilloccurwithdifferentsignsexceeding10minabsolutevalue

1512Two parallel lines are drawn in the plane at the distanceL On thisplaneacircleofradiusRisdroppedThedispersioncenterislocatedatdistanceb outward from one of the parallels Themean deviation of the center of thecircleinthedirectionperpendiculartothisparallelisE

Determineforonethrow(a)theprobabilitythatthecirclewillcoveratleastoneoftheparallels(b)theprobabilitythatitwillcoverbothparallelsifL=10mR=8mb=5mandE=10m

1513Aproduct is considered to be of high quality if the deviation of itsdimensionsfromthestandardsdoesnotexceed345mminabsolutevalueTherandomdeviationsofitsdimensionsobeythenormaldistributionwithamean-squaredeviationof3mmsystematicerrorsareabsentDeterminetheaveragenumberofproductsofhighqualityfromatotaloffouritemsproduced

1514Whatshouldbethewidthofthetolerancefieldinordertoobtainwithaprobabilityatmost00027anitemwhosesizeliesoutsidethetolerancefieldiftherandomdeviationsofthesizefromthemidpointofthetolerancefieldobeythenormaldistributionwithparameters =0andσ=5μ

1515 What should be the distance between two fishing boats sailing onparallel routes so that the probability of sighting a school of fish movingbetweentheboatsinthesamedirectionis05ifthewidthofthestripofsearchforeachboatisanormallydistributedrandomvariablewithparameters =37kmandE=074kmandfordifferentboatsthesequantitiesareindependent

1516Inmanymeasurementsithasbeenestablishedthat75percentoftheerrors (a) do not exceed+ 125mm (b) do not exceed 125mm in absolutevalue Replacing the frequencies of occurrences of the errors by theirprobabilitiesdetermineinbothcasesthemeandeviationofthedistributionlawoftheerrorsAssumethedistributionisnormalwithzeroexpectation

1517TherandomdeviationXofthesizeofanitemfromthestandardobeysthe normal law with the expectation and the mean-square deviation σINondefectiveitemsareconsideredtobethoseforwhichaltXltbTheitems

subjectedtoalterationarethoseforwhichXgtbFind(a)thedistributionfunctionfortherandomdeviationsofthesizesofthe

itemssubjecttoalteration(b)thedistributionfunctionfortherandomdeviationsofthesizesofnondefectiveitems

1518 A normally distributed random variable X has a zero expectationDeterminethemeandeviationEforwhichtheprobabilityP(altXltb)willbelargest(0ltaltb)

16 CHARACTERISTICFUNCTIONS

BasicFormulas

The expectation of the function eiuX (where u is a real variable andiscalledthecharacteristicfunctionE(u)ofarandomvariableX

Foracontinuousrandomvariablewehave

wheref(x)istheprobabilitydensityoftherandomvariableXForadiscreterandomvariable(andonlyforadiscreteone)

inwhichxkaretheparticularvaluesoftherandomvariableandpk=P(X=xk)aretheprobabilitiesthatcorrespondtothem

Ifthemomentmkexiststhen

The probability density f(x) is determined uniquely by the characteristicfunction

Fordiscreterandomvariables the last formulagives theprobabilitydensityin the form of a sum of 8-functions There is a one-to-one correspondencebetweendistributionfunctionsandcharacteristicfunctions


Example161A lot ofn items containsm defective itemsA sample of ritems is drawn from the lot for quality control (m lt r lt n ndash m) Find thecharacteristicfunctionofthenumberofdefectiveitemscontainedinthesample

SOLUTIONTherandomvariableXrepresentingthenumberofdefectiveitemsin thesamplemayassumeall the integralvalueson the interval (0m)Letusdenote

Determiningpk as the ratio between the number of equally probable (unique)mutuallyexclusiveresultsoftheexperimentandthetotalnumberofresultswefind

Consequentlythecharacteristicfunction


Example162Find thecharacteristic functionofa randomvariableXwiththeprobabilitydensity

SOLUTIONSincethecharacteristicfunctionis

thisleadsto

thatis

Problems166to1612canbesolvedinasimilarway

Example163ArandomvariableXhasthecharacteristicfunction

Findtheprobabilitydensityofthisrandomvariable

SOLUTIONTheprobabilitydensityf(x)isrelatedtothecharacteristicfunctionE(u)by

SubstitutingthevalueofE(u)weobtain

Weshallconsideruastherealpartofthecomplexvariablew=u+ivForxlt0theintegralovertherealaxisistheintegraloveraclosedcontour

consistingof therealaxisand thesemicircleldquoof infinite radiusrdquo located in theupperhalf-plane(Figure14)thatis

Bythetheoremofresidues

FIGURE14

ortakingintoaccountthatxlt0wehave

Similarlyforxgt0

wheretheintegrationisextendedoverthesamecontour(Figure14)

Accordingtothetheoremofresidues

orusingthefactthatxgt0wehave

Thereforeforanyvalueofx


Example 164 Find the moments of a random variable X whosecharacteristicfunctionisE(u)=1(1+u2)

SOLUTION Themoments exist up to any order since all the derivatives ofE(u)arecontinuousatoriginConsequently

Weshalldeterminethederivatives

as the coefficients of ukk in the expansion of the function 1(1 + u2) in aMaclaurinseriesthatisweshallusetheequality

Ontheotherhandthefunction1(1+u2)for|u|lt1isthesumofthegeometricprogression

ThustheMaclaurinseriesofthefunction1(1+u2)containsonlyevenpowersofuItfollowsfromthisthat

andthemoments


PROBLEMS

161 Find the characteristic function of the number of occurrences of aneventinonetrialifitsprobabilityofoccurrenceinonetrialisp

162 Find the characteristic function of the number of occurrences of aneventAinnindependenttrialsiftheprobabilityofoccurrenceofAvariesfromonetrialtoanotherandequalspk(k=12hellipn)forthekthtrial

163Determine the characteristic function of a discrete randomvariableXwithabinomialdistributionandalsothecorrespondingM[X]andD[X]

164FindthecharacteristicfunctionofadiscreterandomvariableXobeyingPascalrsquosdistributionlaw

andthecorrespondingM[X]andD[X]165AdiscreterandomvariableXobeysPoissonrsquoslaw

Find(a)thecharacteristicfunctionE(u)and(b)usingE(u) findM[X]andD[X]

166 Find the characteristic function of a normally distributed randomvariablewithexpectation andvarianceσ2

167Findthecharacteristicfunctionandthemomentsofarandomvariablewiththeprobabilitydensity

168 Find the characteristic function and all the moments of a randomvariableuniformlydistributedovertheinterval(ab)


Finditscharacteristicfunction1610ArandomvariableXhastheprobabiltydensity

Finditscharacteristicfunctionandmoments1611 Find the characteristic function of a random variable X whose

probabilitydensity(thearcsinelaw)is

1612 Find the characteristic function of a random variable X obeyingCauchyrsquosdistributionlaw

1613Usingtheexpression

for the characteristic function of the normal distribution law determine thecharacteristic function of the random variable (a) Y = aX + b (b)


for the characteristic function of a centralized randomvariableX that obeys anormaldistributionlawdetermineallitscentralmoments

1615ThecharacteristicfunctionofarandomvariableXisgivenintheform

DeterminetheprobabilitydensityofX1616Giventhecharacteristicfunctions

determinethecorrespondingprobabilitydensities1617Giventhecharacteristicfunction

show that it corresponds to a discrete random variable Find the distributionseriesofthisvariable

17 THECOMPUTATIONOFTHETOTALPROBABILITYANDTHEPROBABILITYDENSITYINTERMSOFCONDITIONALPROBABILITY

BasicFormulasThetotalprobabilityofaneventAisgivenbytheformula

inwhichf(x)istheprobabilitydensityoftherandomvariableXonthevaluesofwhichdepends the probability of occurrence ofAP(A|x) is the probability ofoccurrence of the event A computed under the assumption that the random

variablexassumesthevaluexTheconditionalprobabilitydensityf(x |A)ofarandomvariableX ie the

probabilitydensityunder theassumption thatAoccurred isdeterminedby theformula(thegeneralizedBayesformula)

inwhich f(x) is theprobabilitydensityprior to the experiment of the randomvariableX


Example171TheprobabilityofaneventdependsontherandomvariableXandcanbeexpressedbythefollowingformula

FindthetotalprobabilityoftheeventAifXisanormallydistributedrandomvariablewithexpectation andvarianceσ2

SOLUTIONThetotalprobabilityoftheeventAis

Substitutingherethegivenprobabilitydensity

weobtain

Theexponentofeinthelastintegralcanbereducedtotheform

Consequently

Since

then


Example172Thedeviationofthesizeofanitemfromthemidpointofthetolerancefieldofwidth2dequalsthesumoftworandomvariablesXandYwithprobabilitydensities

and

Determinethe(conditional)probabilitydensityoftherandomvariableXforthe nondefective items if the distribution φ(y) does not depend on the valueassumedbyX

SOLUTION Let A denote the event that an item produced turns out to benondefectiveTheconditionalprobabilityP(A|x)ofgettinganondefectiveitemundertheassumptionthattherandomvariableXtakesthevaluexis

Letf(x|A)betheconditionalprobabilitydensityofXfornondefectiveitemssothat

Substitutingthevaluesoff(x)andP(A|x)weobtain

or

PROBLEMS

171Supposethatastraightlineisdrawnintheplaneandonitaremarkedpoints separated by the distance l Determine the probability that at least onepointwillcoincidewith thecenterofacircleofdiameterbandmoving in thesameplanesothatitscenterdescribesastraightlineintersectingthegivenlineatanangleθequallyprobableover the interval (θθ)Theanglesandθ1andθ2satisfytheconditionssinθ1ltblandsinθ2gtbl)

172 On each of two parallel lines points are taken independently at aconstantintervall=100mDeterminetheprobabilitythatatleastonepointwilllieinaninfinitestripofwidthD=25mlocatedinthesameplaneasthetwoparallelssothatthelinesthatbounditareperpendiculartotheseparallels

173Findtheprobabilityofhittingatargetinonetrialifthedistancetothetargetat the instantof theshot isarandomvariableuniformlydistributedovertheinterval100to200mandtheconditionalprobabilityofhittingthetargetis3000D2whereDisexpressedinmeters

174OnashoreofabayofwidthL=30kmthereisanobservationstationwhosedistanceofobservationisanormallydistributedrandomvariablewiththeexpectation = 20 km andmean deviationE = 1 kmA ship can passwithequalprobabilitythroughthebaywhilemovingalongtheshoreatanydistancefrom thestationFind theprobability that theobservationstationwilldiscovertheship

175On the rightpanof abalance a load is placedwhosemassobeys thenormaldistributionlawwithparameters =20kgandE=1kgOntheleftpananotherloadisplacedwhosemassisequallyprobablewithinthebounds0to50kg Determine the probability that the right pan will outweigh the left oneComparetheresultwiththatobtainedundertheassumptionthattheloadontherightpanisnotrandombutisexactly20kg

176ConsideranumbernofindependentmeasurementsofanormalrandomvariableXwhoseexpectationcoincideswith theoriginof thereferencesystem

andwithmeandeviationR Find theprobability that the result of at least onemeasurement will deviate from the random variable Z by at most plusmnr if Z isuniformlydistributedovertheinterval(ndashll)

177 Given a sequence of random variables X1 X2 Xn with the sameprobabilitydensityf(x)wecalltherandomvariable

inwhichXmaxisthemaximumandXmintheminimumoftheobtainedvaluesXj(j=12hellipn)therange

Findthedistributionfunctionoftherange

178Whatistheprobabilitythattwopointsselectedrandomlyinacirclewilllieononesideofachordparalleltoagivendirectionandwhosedistancefromthecenterisauniformlydistributedrandomvariable

179 The coordinates Xi of the random points A1 A2 hellip An have theprobabilitydensities

OneofthesenpointscoincideswithapointA0whosedeviationofcoordinatesfromagivennumberhastheprobabilitydensityf(x)DeterminetheprobabilitythatthepointAwillcoincidewithA0

1710ArandomvariableXobeysPoissonrsquoslaw

whoseparameterisunknownbutpriortotheexperimenttheparameterhastheprobabilitydensity

After theexperiment a randomvariableX assumes thevaluem0 Find theprobabilitydensityaaftertheexperiment

1SeeforexampleGelrsquofandIMandShilovGEGeneralizedFunctionsVol1Propertiesand

OperationsTranslatedbyESaletanNewYorkAcademicPressInc19642IgnorescatteringandabsorpotionofparticlesAvagadrorsquosnumberN0=602times10

23isthenumberofatomsinaquantityofatomsinaquantityofthesubstancewhosemassingramsequalsitsatomicweightThehalf-timeTpisthetimeduringwhichamassofsubstancedecaystohalftheoriginalmass


18 DISTRIBUTIONLAWSANDNUMERICALCHARACTERISTICSOFSYSTEMSOFRANDOMVARIABLES

BasicFormulasThe distribution function (integral distribution law) F(x1 x2 hellip xn) of a

system of n random variables (X1 X2 hellip Xn) is defined by the formula

For a systemof continuous randomvariables there can exist a probabilitydensity (differential distribution law) defined by the formula

A system of discrete random variables is characterized by the set ofprobabilitiesP(X1=i1X2=i2hellipXn=in)whichcanbereducedtoatablewithnrows(accordingtothenumberofrandomvariables)

Thedistributionfunctionforcontinuousrandomvariablescanbeexpressedintheformofamultipleintegral

andfordiscreterandomvariablesintheformofthemultiplesum

inwhichthesummationisextendedoverall thepossiblevaluesofeachof therandomvariablesforwhichi1ltx1i2ltx2hellipinltxn

Forn=2asystemofcontinuousrandomvariablescanbeinterpretedasarandompointintheplaneandforn=3asarandompointinspace

The probability that a random point lands in a region S is obtained byintegratingtheprobabilitydensityoverthisregion

Thebasicnumericalcharacteristicsofasystemofn randomvariablesaretheexpectations

thevariances

andthecovariances

Themomentsfordiscreterandomvariablescanbecomputedsimilarlyietheintegrationisreplacedbysummationoverallpossiblevaluesoftherandomvariables

Thesecondcentralmomentsformthecovariancematrix

inwhichkij=kjiSometimesitisveryconvenienttousetheformula

The random variables X1 X2 hellip Xn are said to be uncorrelated if thenondiagonalelementsofthecovariancematrixarezero

Thenondimensionalcorrelationcharacteristicbetweentherandomvariables

XiandXjisthecorrelationcoefficent

Thecorrelationcoefficientsformthenormalizedcovariancematrix

inwhichrij=rjiThecontinuousrandomvariablesX1X2hellipXnformingasystemarecalled

independentif

andarecalleddependentif

wherefi(xi)istheprobabilitydensityoftherandomvariableXi(seeSection20)Thediscrete randomvariablesX1X2hellipXn are said tobe independent if


Example181Asa resultofa testan itemcanbeclassifiedas firstgradewiththeprobabilityp1secondgradewiththeprobabilityp2ordefectivewiththeprobability p3 = 1 ndash p1 ndashp2 A number of n items are tested Determine theprobabilitydensityfordifferentnumbersofitemsoffirstandsecondgradetheirexpectationsvariancesandcovariances

SOLUTIONLetXdenotethenumberofitemsoffirstgradeandYofsecondgrade Since the tests are independent the probability that k items will beclassified as first grade s items as second grade and the remainingn ndash k ndash sitemsasdefective(takingintoaccountallthepossiblecombinationsofthethreeterms k s and n ndash k ndash s of which the sum is composed) is

Thevaluesofthisprobabilityfork=01hellipns=01hellipnandk+slenformtherequiredsetofprobabilitiesfordifferentnumbersofitemsoffirstandsecond grade The expectation of the number of first grade items is

Thevarianceofthenumberoffirstgradeitemsis

Similarlywefindthat

Thecovariancebetweenthenumberoffirstgradeandsecondgradeitemsis

Example182For theprobabilitydensityof a systemof randomvariables(X Y)

determine (a) thedistribution functionof thesystem (b) theexpectationsofXandY(c)thecovariancematrix

SOLUTIONWefirstfindthedistributionfunction(for0lexleπ2and0leyleπ2)

TheexpectationoftherandomvariableXis

ThevarianceofXis

FromthesymmetryoftheprobabilitydensityaboutXandYitfollowsthat

FIGURE15

Thecovarianceis

Thereforethecovariancematrixhastheform

InasimilarwayProblems1818and1819canbesolved

Example183Aneedleoflengthlisdroppedonasmoothtableruledwithequidistantparallel linesatdistanceLapartDetermine theprobability that theneedlewillcrossoneofthelinesiflltL(Buffonrsquosproblem)

SOLUTION Introduce a system of random variables (X Φ)whereX is thedistancefromthemidpointof theneedle to thenearest lineandΦis theacuteanglemadebytheneedlewiththisline(Figure15)ObviouslyXcanassumeallvaluesfrom0toL2andΦfrom0toπ2withequalprobabilityThereforef(xφ)=2L2πL=4πLfor0lexleL20leφleπ2

Theneedlewill crossoneof the lines for a givenφ if 0lex le (l sinφ)2

Fromthisitfollowsthat


PROBLEMS

181 The coordinates X Y of a randomly selected point are uniformlydistributed over a rectangle bounded by the abscissas x = a x = b and theordinates y = c y = d (b gt a d gt c) Find the probability density and thedistributionfunctionoftherandomvariables(XY)

182 A system of random variables (X Y) has the probability density

Determine(a)themagnitudeofA(b)thedistributionfunctionF(xy)183Determinetheprobabilitydensityofasystemofthreepositiverandom

variables (X Y Z) if their distribution function is

184 Under the assumptions of the preceding problem find the locus ofpointswiththesameprobabilitydensity

185Fromasampleofn=6itemsXturnouttobenondefectiveandoftheseY(Y le3) areof excellent qualityThe system (XY) is given by the followingtwo-dimensional probability distribution table (matrix) (Table7) (a) Form thedistributionfunction (b) find theprobabilityofobtainingat least two itemsofexcellentquality(c)findM[X]M[Y]andthecovariancematrix

TABLE7

186AsystemofindependentrandomvariablesX1X2hellipXnisgivenbytheprobabilitydensitiesf1(x1)f12(x2)hellipfn(xn)Determinethedistributionfunctionofthissystem

187TheprobabilitydensityofasystemoftworandomvariablesX1andX2that can bemeasured only simultaneously is f(x1x2) The values u and v areobservedFindtheprobabilitythatuwillbethevalueoftherandomvariableX1andvthatofX2

188 Assume that the probability density for a system of three randomvariablesthatcanbemeasuredonlysimultaneouslyisf(x1x2x3)Thevaluesofu vw are observed but it is not known how these values and the random

variablescorrespondDeterminetheprobabilitythatuistherealizationofX1andwthatofX3

189 Find the probability that a randomly selected point is located in theshadedregionshowninFigure16ifthedistributionfunctionF(xy)isknown

FIGURE16

1810 What is the probability that a point with coordinates (X Y) hits aregion specified by the inequalities (1 le x le 2 1 le y le 2) if the distributionfunction (a gt 0)

1811 The coordinates of a random point (XY) are uniformly distributedoverarectangleboundedbytheabscissas0andaandordinates0andbFindtheprobabilitythatarandompointhitsacircleofradiusRifagtbandthecenterofthecirclecoincideswiththeoriginofthecoordinates

1812 The probability density of a system of random variables is

Find(a)theconstantc(b)theprobabilityofhittingacircleofradiusaltRifthecentersofbothcirclescoincidewiththeorigin

1813TherandomvariablesXandYarerelatedbytheequalitymX+nY=cinwhichmnandcareconstants(mne0nne0)

Find (a) the correlation coefficient rxy (b) thequotient of themean-squaredeviationsσxσy

1814 Prove that the absolute value of the correlation coefficient does notexceedone

1815Showthat

1816Suppose that the covariancematrix of a systemof randomvariables

(X1X2X3)is

Formthenormalizedcovariancematrix||riy||1817Someitemsareclassifiedbytheirshapeasroundorovalandbytheir

weightaslightorheavyTheprobabilitiesthatarandomlyselecteditemwillberoundandlightovalandlightroundandheavyorovalandheavyareαβγandδ=1ndashαndashσrespectivelyFindtheexpectationsandvariancesforthenumberXof round items andY of light items and also the covariance kxy between thenumberofrounditemsandlightitemsifα=040β=005γ=010

1818Determine theexpectationsandthecovariancematrixofasystemofrandom variables (X Y) if the probability density is

1819Findtheprobabilitydensitytheexpectationandthecovariancematrixofasystemofrandomvariables(XY)definedontheintervals(0lexleπ2)and(0 le y le π2) if the distribution function of the system is

1820SolveBuffonrsquosproblem ie find theprobability that theneedlewillcrossatleastoneofthelinesinthecaselgtL(seeExample183)

1821Aneedleoflengthl isdroppedonaplanepartitionedintorectangleswithsidesaandbDeterminetheprobabilitythat theneedlewillcrossat leastonesideofarectangleifaltlbltl

19 THENORMALDISTRIBUTIONLAWINTHEPLANEANDINSPACETHEMULTIDIMENSIONALNORMALDISTRIBUTION

BasicFormulas

Theprobabilitydensityofasystemoftwonormalrandomvariables(XY)is(for a normal distribution of the coordinates of a point in the plane)

where are the expectations of X and Y σx σy are the mean-squaredeviationsandristhecorrelationcoefficientofXwithY

The locus of pointswith equal probability density is an ellipse (dispersionellipse)definedbytheequation

Ifr=0thenthesymmetryaxesofthedispersionellipseareparalleltothecoordinateaxesOxandOytherandomvariablesXandYareuncorrelatedandindependent and the probability density is

where arethemeandeviationsofXandYrespectivelyandρ=04769hellip

Theellipsedefinedbytheequality

iscalledtheunitellipseThe probability density of a system of n normal random variables (for a

multidimensionalnormaldistribution)is

where

is thedeterminantformedbytheelementsof thecovariancematrix are

theelementsoftheinversematrixandAijisthecofactoroftheelementkij

In thecaseof three independentnormalrandomvariablesXYZwehavekxy = kyz = kxz = 0 and

whereExEyEzarethemeandeviationsofXYZrespectivelyThisisaparticularcasewherethesymmetryaxesoftheellipsoidareparallel

tothecoordinateaxesOxOyandOz


Example 191 Given the covariance matrix of a system of four normal

randomvariables(X1X2X3X4)determinetheprobabilitydensityf(x1x2x3x4if =10 =0 =ndash10=1

SOLUTIONWefirstcomputethecofactorsofthedeterminantΔ=|kij|

Nextwefindthevalueofthedeterminant

Inderivingtheformulafortheprobabilitydensitywetakeintoaccountthefact that for i ne j the exponent contains equal terms

Theprobabilitydensityis

Example 192 A random point in space is given by three rectangularcoordinates forminga systemofnormal randomvariableswith theprobabilitydensity

(a)Find the covariancematrix (b)determine the locusofpointswhen theprobabilityis001

SOLUTION(a)Since

where

then

Thisimpliesthat

Forverificationwecancomputethenormalizationfactor

(b) The required locus of points with constant probability density is thesurfaceoftheellipsoid

Example193 Find theprobability that apoint (XYZ) lands in a regionrepresentingahollowparallelepipedwhoseoutersurfaceisgivenbytheplanes

andwhoseinnersurfaceisgivenbytheplanes

The dispersion of points (X Y Z) obeys a normal distribution with theprincipalaxesparalleltothecoordinateaxesthedispersioncenteratthepoint

andmeandeviationsExEyEz

SOLUTIONSince theprincipaldispersionaxesareparallel to thecoordinateaxestheeventthatoneofthecoordinatesforinstancexwillassumevaluesontheinterval(ab)isindependentofthevaluesassumedbytheothercoordinatesTherefore

inwhich

TheprobabilitiesoftheotherinequalitiescanbedeterminedsimilarlyTherequiredprobabilityofreachingtheinteriorofthehollowparallelepiped

will be determined as the difference between the probabilities of reaching theparallelepipeds bounded by the outer and inner surfaces ie

PROBLEMS

191ItisknownthatXandYareindependentnormalrandomvariableswithexpectations and andmeandeviationsExandEyrespectivelyExpressthedistribution function of the system (XY) in terms of the normalized Laplacefunctions

192 Given the expectations of two normal random variablesM[X] = 26

M[Y]=ndash12andtheircovariancematrixdeterminetheprobabilitydensityofthesystem(XY)193Given theprobabilitydensityfor thecoordinatesofarandompoint in

the plane

find(a)constantc(b)thecovariancematrix(c)theareaSeloftheunitellipse194Determineatthepointx1=2x2=2theprobabilitydensityofasystem

of two normal random variables for which and

195 Given thecovariance matrix of a system of three normal random

variables(XYZ)andexpectations findtheprobabilitydensityf(xyz)anditsmaximumvalue

196 A system of n normal random variables has the covariance matrix

(a)Compute the inverseof thismatrix (b) find theprobability f(x1x2hellipxn)if

197Thecoordinates(X1Y1)and(X2Y2)oftworandompointsintheplaneobey the normal distribution lawwith the expectations of all coordinates zeroand the variances of all coordinates equal to 10 The covariances betweencoordinates with the same symbol are equalM[X1X2] =M[Y1 Y2] = 2 theremainingpairsofcoordinatesareuncorrelatedFindtheprobabilitydensityf(x1y1x2y2)

198Thecoordinates(XY)ofarandompointAintheplaneobeythenormal

lawDetermine the probability that A will turn out to be inside an ellipse with

principalsemi-axeskaandkb andcoincidingwith thecoordinateaxesOx andOy

199Thecoordinates(XYZ)ofarandompointAinspaceobeythenormaldistribution law

FindtheprobabilitythatAisinsideanellipsoidwiththeprincipalsemi-axeskE1kE2andkE3coincidingwiththecoordinateaxesOxOyandOz

1910 The determination of the coordinates of a point in the plane isaccompaniedbyasystematicerrordinoneofitsrectangularcoordinatesandarandomerrorobeyingacircularnormaldistributionwithmeandeviationEFindtheprobabilitythatthedeviationofthepointfromitsmeasuredpositionwillnotexceedaquantityR

1911Asystemofrandomvariables(XY)obeysanormaldistributionwithnumericalcharacteristicsM[X]=M[Y]=0Ex=Ey=10kxy=0Determinetheprobabilitythat(a)XltY(b)Xgt0Ylt0

1912ComputetheprobabilitythatarandompointAwithcoordinatesXYand obeying a normal distribution lawwill lie in a rectanglewhose sides areparallel totheprincipaldispersionaxesif thecoordinatesofitsverticesare(ab)(ad)(cb)(cd)fora=ndash5b=10c=5d=20and =0 =10Ex=20Ey=10

1913ArandompointisdistributedinaccordancewithanormalcircularlawwithmeandeviationE=10mComparetheprobabilityofhittingafigurewhoseareais314sqmifitsshapeis(a)acircle(b)asquare(c)arectanglewhosesides are in the ratio 101The dispersion center coincideswith the geometriccenterofthisfigure

1914 Find the probability that a randomly selected point lies inside theshaded region (Figure 17) bounded by three concentric circles and the raysissuingfromtheircommoncenteriftheradiusoftheexteriorcircleisRandthedispersionofthepointintheplaneobeysacircularnormaldistributionlawwithmeandeviationEThedispersioncentercoincideswiththecenterofthecircles

FIGURE17

1915Findtheprobabilityofhittingafigureboundedbythearcsdeterminedby the radiiR1 andR2 and the rays issuing from the common centerO if thedispersion of a randompoint in the plane obeys a circular normal distributionwithmean deviationE and the angle made by the rays is α The dispersioncentercoincideswithM(R1ltR2)

1916Theprobabilityofhittingarectanglewithsides2dand2kandparallelto the principal dispersion axes satisfies the following approximate formula

whichisrecommendedwhendExandkEzdonotexceed15Equatingthezeroand secondmoments on the left-and right-hand sides of this equality find thevaluesofAαβ

1917Usingtheapproximateformulafromtheprecedingproblemfind theprobabilityofhitting a rectanglewith sides2d and2k parallel to theprincipaldispersion axes if the coordinates of the dispersion center are uniformlydistributedover thegiven rectangle andExEz are knownCompare the resultobtainedwiththeprobabilityofadirecthitinthesameregionwhenthecenterofdispersioncoincideswiththecenteroftheregion

1918Atargetconsistsoffourconcentriccirclesofradii102030and40

cmrespectively(Figure18)Byhittingthebullrsquos-eyeonescores5pointsandforeachofthethreeannulimdash43and2pointsThescoreissatisfactoryifonescoresatleast7pointsinthreeshotsandexcellentifonescoresmorethan12points What is the probability of a satisfactory score in the case of circularnormal dispersionwithmean deviation 20 cmWhat is the probability of anexcellentscoreThedispersioncentercoincideswiththecenterofthetarget

1919WhatistheprobabilityofhittingarighttriangleABCwithlegsBC=aandAC=bparalleltotheprincipaldispersionaxes(AC||Oy)(BC||Ox)ifthe

dispersioncentercoincideswithpointAand

FIGURE18

1920 Find the probability that a pointwith coordinatesXYZ will hit aregionrepresentingasphereofradiusRfromwhichacentralcubewithedgeahasbeenremoved(thediagonalof thecube isshorter than thediameterof thesphere)Thedispersioncentercoincideswith thecommoncenterof thesphereandthecubeThedistributionisnormalsphericalwithmeandeviationE

1921 Find the probability that a point A(X Y Z) will lie inside a rightcylinderwhosebasehasradiusRandwhoseheightishifthedispersioninthexy-plane parallel to the base obeys a normal circular distribution with meandeviationE and thedispersionalong thegenerator is independentofXYandobeys(a)anormaldistributionwithmeandeviationB (thedispersioncenterislocatedontheaxisofthecylinderanddividesitintheratiomn)(b)auniformdistributionovertheinterval(ndashHH)forHgth

1922FindtheprobabilitythatarandompointA(XYZ)will lie ina rightcircularconewhosevertexcoincideswiththedispersioncenterwhoseheightis

handwhosebasehasradiusRthedispersioninthexy-planewhichisparallelto the base obeys a normal circular law with mean deviation E and thedispersion along the height is independent of X Y and obeys a normaldistributionwithmeandeviationa

1923Anormaldistributionlawintheplaneisgivenbytheexpectationsofrandom variables and the covariance matrix

Findthelocusofpointswithprobabilitydensity10ndash51924 A normal distribution law in space is given by the expectations

and the covariance matrix

Findthelocusofpointswhoseprobabilitydensityis10ndash51925For themultidimensionalnormaldistributiongiven inProblem196

find the locusof pointswithprobability density 10ndash5Find thevalueofn forwhichthisproblemhasnosolutions

20 DISTRIBUTIONLAWSOFSUBSYSTEMSOFCONTINUOUSRANDOMVARIABLESANDCONDITIONALDISTRIBUTIONLAWS

BasicFormulasIfF(xy) is thedistribution functionof a systemof two randomvariables

then the distribution function of the random variable X is

SimilarlythedistributionfunctionofYis

Theprobabilitydensitiesoftherandomvariablescontainedinthesystemare

If F(x1 x2 hellip xn) is the distribution function of a system of n randomvariables thenthedistributionfunctionofsomeof thesevariables(subsystemsof random variables) for example X1 X2 hellip Xk is

andthecorrespondingprobabilitydensityis

Theprobabilitydensityofoneoftworandomvariablescomputedundertheassumption that the other random variable assumes a certain value (theconditional probability density) is

Theprobabilitydensityofthesubsystemoftherandomvariables(X1X2hellipXk)computedundertheassumptionthattheremainingrandomvariablesXk+1Xk + 2 hellip Xn assume certain values is

The probability density of a system can be expressed in terms of theconditionaldensitiesbytheformula


Example201ThepositionofarandompointA(XY)isequallyprobableat

anypointofanellipsewiththeprincipalsemi-axesaandbcoincidingwiththecoordinateaxesOxandOyrespectively

(a) Determine the probability density of each of the two rectangularcoordinates and their mutual conditional probability densities (b) analyze thedependenceandthecorrelationoftherandomvariableformingthesystem

SOLUTION(a)Since

foragivenxontheinterval(ndashaa) theprobabilitydensity f(xy)differs from

zeroonlyif thisimpliesthat

For|x|gtafx(x)=0Fromthisweobtain

Similarly

and

(b)ThecovariancebetweenXandYis

wherethefunctionintegratedisdifferentfromzeroinsidetheellipse

Makingthechangeofvariables

weobtain

ThustherandomvariablesXandYareuncorrelated(kxy=0)butdependent

since

Example 202 The coordinates of a random point in the plane obey thenormaldistributionlaw

Determine (a) the probability density of the coordinates X and Y (b) theconditionaldensitiesf(y|x)andf(x|y)(c)theconditionalexpectations(d)theconditionalvariances

SOLUTION(a)FortheprobabilitydensityofthecoordinateXwefind


andconsideringthefactthat

weobtain

or

Similarlywefindthat

(b)Dividingf(xy)byfx(x)weobtain

andsimilarly

(c)FromtheexpressionsforconditionalprobabilitydensitiesitfollowsthattheconditionalexpectationoftherandomvariableYforafixedvalueX=x is

Similarly

These equations expressing the linear dependence of the conditionalexpectationofoneoftherandomvariablesonafixedvalueoftheothervariablearecalledtheregressionequations

(d)Fromtheexpressionsforconditionaldistributiondensitiesitfollowsthattheconditionalvariancesare

Example203Determine the probability density of the length of a radius-vectorif thecoordinatesofitsendAobeythenormalcirculardistributionlaw

SOLUTIONWepassnowfrom the rectangularcoordinatesofA to thepolarcoordinates(rφ)Theprobability that theradius-vectorassumesvaluesontheinterval (r r + dr) is approximately fr(r) dr and can be interpreted as theprobabilityforarandompointAtolieinaninfinitelynarrowannulusshowninFigure19

Consequently

FIGURE19

Integratingwithrespecttothevariablesrφandconsideringtheexpressionfor f(x y) we obtain

(Rayleighrsquosdistribution)

PROBLEMS

201Asystemofrandomvariables(XYZ)isuniformlydistributedinsidearectangularparallelepipeddeterminedbytheplanesx=a1x=a2y=b1y=b2z = c1 z = c2 Find the probability densities of the system (X Y Z) of thesubsystem (YZ) and of the random variableZ Verify the dependence of therandomvariablesformingthesystem

202Thepositionofarandompoint(XY)isequallyprobableanywhereonacircleof radiusR andwhose center is at theoriginDetermine theprobabilitydensityandthedistributionfunctionofeachoftherectangularcoordinatesArerandomvariablesXandYdependent

203 Under the assumption made in the preceding problem find theprobabilitydensityf(y|x)for|x|ltR|x|=Rand|x|gtR

204UndertheassumptionsofProblem202computethecovariancematrixofthesystemofvariablesXandYArethesevariablescorrelated

205A system of random variablesXY obeys a uniform distribution lawover a square with side a The diagonals of the square coincide with thecoordinateaxes

(a) Find the probability density of the system (X y) (b) determine theprobability density for each of the rectangular coordinates (c) find the

conditional probability densities (d) compute the covariance matrix of thesystemofrandomvariables(Xy)(e)verifytheirdependenceandcorrelation

206Therandomvariables(XYZ)areuniformlydistributedinsideasphereofradiusRDetermineforpointslyinginsidethisspheretheprobabilitydensityofthecoordinateZandtheconditionalprobabilitydensityf(xy|z)

207 Given the differential distribution law for a system of nonnegativerandomvariablesdeterminekfx(x)fy(y)f(x|y)f(y|x)andthefirstandsecondmomentsofthedistribution

208Givenfy(y)M[X|y]andD[X|y]forasystemofrandomvariables(XY)findM[X]andD[X]

209Asystemoftworandomvariables(XY)obeysthenormaldistributionlaw

Determine(a)theconditionalexpectationsandvariances(b)theprobabilitydensityofeachoftherandomvariablesformingthesystem(c)theconditionalprobabilitydensitiesf(y|x)andf(x|y)

2010Theprobabilitydensityofasystemoftworandomvariables(XY) isgivenintheform

Findthedistributionlawfx(x)andfy(y)UnderwhatconditionsareXandYindependentrandomvariables

2011Given the probability density of a system of two randomvariables

find the constant k the covariance between X and Y and the conditionaldistributions f(x |y) and f(y |x)2012Thepositionof a referencepoint in theplaneisdistributedaccordingtoanormallawwith =125m =ndash30mσx=40mσy=40mandrxy=06ThecoordinateXdefinesthedeviationof thereferencepointwith respect to the ldquodistance rdquo iewith respect to a directionparallel to the observation line The coordinateY defines the deviation of thereference point with respect to a lateral ldquodirectionrdquo perpendicular to theobservationlineThedeviationsareestimatedfromtheoriginofcoordinates

Determine(a)theprobabilitydensityofthedeviationsofthereferencepoint

withrespect to thedistance(b) theprobabilitydensityof thedeviationsof thereference point with respect to the lateral direction (c) the conditionalprobability density of the deviations of the reference point with respect todistanceinabsenceof lateraldeviations(d) theconditionalprobabilitydensityof the deviations of the reference point with respect to lateral direction for adeviationwithrespecttothedistanceequalto+25m

2013Under theassumptionsof theprecedingproblem find the regressionequationsofYonXandXonY

2014Determinetheprobabilitydensityofthelengthoftheradius-vectorforarandompointanditsexpectationifthecoordinates(XYZ)ofthispointobeythe normal distribution law

2015ThecoordinatesofarandompointAinthexy-planeobeythenormal

distributionlaw

Findtheprobabilitydensitiesfr(r)andfφ(φ)forthepolarcoordinatesofthispoint

2016Undertheassumptionsoftheprecedingproblemfindtheconditionalprobabilitydensitiesf(r|φ)andf(φ|r)

2017 A random point in space obeys the normal distribution law

Find(a)theprobabilitydensityofthesphericalcoordinatesofthispoint(RΘΦ) if x = r cos θ cosφ y = r cos θ sinφ z = r sin θ (b) the probabilitydensities of the subsystems (R Θ) and (ΘΦ) (c) the conditional probabilitydensitiesf(r|θφ)andf(φ|rθ)

2018For the systemof randomvariablesX1Y1X2Y2 of Problem 197findtheprobabilitydensitiesofthesubsystemsfx1x2(x1x2)andfx1y1(x1y1)

2019 Under the assumptions of the preceding problem determine theprobability density f(x2 y2 | x1 y1) the conditional expectations and theconditional variances


21 NUMERICALCHARACTERISTICSOFFUNCTIONSOFRANDOMVARIABLES

BasicFormulasTheexpectationandvarianceofarandomvariableYthatisagivenfunction

Y =φ(X) of a random variableX whose probability density f(x) is known isgivenbytheformulas

In a similar way onemay find themoments and centralmoments of anyorder

The foregoing formulasextend toanynumberof randomarguments ifY=φ(X1X2hellipXn)then

where f(x1 x2 hellip xn) is the probability density of the system of randomvariables(X1X2hellipXn)

For discrete random variables the integrals in the preceding formulas arereplacedbysumsandthedensitiesbyprobabilitiesofthecorrespondingsetsofvaluesofX1X2hellipXn

Ifthefunctionφ(X1X2hellipXn)islinearthatis

then

wherekijisthecovariancebetweentherandomvariablesXiandXjKnowledge of the distribution law of the random arguments for the

determinationofthemomentsofthefunctionisunnecessaryinsomecasesLetZ=XYthenM[Z]=M[X]M[Y]+kxyFurthermoreifXandYareuncorrelatediethecovariancekxyvanishesthen

Thelastformulacanbegeneralizedforanynumberofindependentrandomvariables

Ifthemomentsofthelinearfunction

ofindependentrandomvariablesexisttheycanbedeterminedbytheformula

where is the characteristic function of therandomvariableXj

Theasymmetrycoefficientand theexcessof the randomvariableY in thiscasearegivenbytheformulas

where


Example211ArandomvariableXobeysabinomialdistributionlawFindtheexpectationandvarianceoftherandomvariableY=eaX

SOLUTION The random variable X can assume values 0 1 2hellip n Theprobability that it will assume the value m is determined by the formula

Therefore

Example212ThescreenofanavigationalradarstationrepresentsacircleofradiusaAsaresultofnoiseaspotmayappearwithitscenteratanypointofthecircleFindtheexpectationandvarianceofthedistancebetweenthecenterofthespotandthecenterofthecircle

SOLUTIONThe randomdistanceR from the center of the circle to the spotcanbeexpressedintermsofrectangularcoordinatesXandYas

Theprobabilitydensityof the systemof randomvariables (XY) is knownandisgivenbytheformula

Therefore

Inamannersimilar to thatused inExamples211and212onecansolveProblems211to21142120to2124212621272129and2130

Example213AsampleofnitemsisdrawnwithoutreplacementfromalotofNitemsofwhichT=NparedefectiveFindtheexpectationandvarianceofthenumberofdefectiveitemsinthesample

SOLUTIONLetXdenotetherandomnumberofdefectiveitemsinthesampleThe randomvariableX canbe represented as where the

randomvariableXjequals1ifthejthitemselectedturnsouttobedefectiveandzerootherwiseTheprobabilityisp thatthevalueis1andconsequently =M[Xj]0middot(1ndashp)+1middotp=p(asinExample61onecanshowthattheprobabilityofobtainingadefectiveitemdoesnotdependonj)

Then

If sampling is done without replacement the random variables Xj aredependentandhence

where

Finally


Example214 Find the expectation for the square of the distance betweentwopointsselectedatrandomontheboundaryofarectangle

SOLUTIONByselecting two randompointson theboundaryofa rectanglethe following unique mutually exclusive events (hypotheses) may occur (seeFigure20)H1thatthepointslieonthesamesideaH2thatthepointslieonthesamesidebH3 that thepoints lie on adjacent sidesH4 that the points lie onoppositesidesaH5thatthepointslieonoppositesidesb

Fortheprobabilitiesofthesehypotheseswehave

where2pistheperimeteroftherectangle

FIGURE20

Determine the conditional expectation (ie the expectation with theassumptionthatthehypothesisHioccurs)forthesquareofthedistancebetweentwopoints

WefindthatthetotalexpectationsoftherandomvariableZ2is

Problems2118and2119canbesolvedsimilarly

PROBLEMS

211Findtheexpectationofthelengthofachordjoiningagivenpointonacircleofradiusawithanarbitrarypointonthecircle

212Findtheexpectationofthelengthofachorddrawninacircleofradiusaperpendiculartoachosendiameterandcrossingitatanarbitrarypoint

213Somesteelballsaresortedaccordingtotheirsizesothatthegroupwithratedsize10mmcontainsballsthatpassthroughacircularslotof101mmanddonotpassthroughaslotofdiameter99mmTheballsaremadeofsteelwithspecificweight78gccFind theexpectationandvarianceof theweightofaballbelongingtoagivengroupifthedistributionoftheradiusinthetolerancerangeisuniform

214 A fixed pointO is located at altitude h above the endpoint A of a

horizontalsegmentAKoflengthlApointB israndomlyselectedonAKFindtheexpectationoftheanglebetweensegmentsOAandOB

215Thelegsofacompasseach10cmlongmakearandomangleφwhosevaluesareuniformlydistributedovertheinterval[0180deg]Findtheexpectationofthedistancebetweentheendsofthelegs

216 A random variable X obeys a normal distribution law Find theexpectationoftherandomvariableYif

217 The vertex C of the right angle of an isosceles right triangle isconnectedbyasegmentwithanarbitrarypointMonthebasethelengthofthebaseis2mFindtheexpectationofthelengthofsegmentCM

218ApointisselectedatrandomonacircumferenceofradiusaFindtheexpectationoftheareaofasquarewhosesideequalstheabscissaofthispoint

219An urn contains white and black balls The probability of drawing awhiteballispanddrawingablackoneqAnumbernofballsaredrawnonebyone with replacementWhat is the expectation of the number of instances inwhichawhiteballfollowsablackone

2110AsystemofrandomvariablesXYobeysthenormaldistributionlaw

Findtheexpectationoftherandomvariable

2111TwopointsXandYarerandomlyselectedinasemicircleofradiusaThesepointsandoneendoftheboundingdiameterformatriangleWhatistheexpectationoftheareaofthistriangle

2112ThreepointsABandCareplacedat randomonacircumferenceofunitradiusFindtheexpectationoftheareaofthetriangleABC

2113ThenumberofcosmicparticlesreachingagivenareaintimetobeysPoissonrsquoslaw

TheenergyofaparticleisarandomvariablecharacterizedbyameanvalueFindtheaverageenergygainedbytheareaperunittime

2114Anelectronic systemcontainsn elementsThe probability of failure(damage)ofthekthelement ispk (k=12hellipn)Find theexpectationof thenumberofdamagedelements

2115Asystemconsistingofnidenticalunitsstopsoperatingifatleastoneunit fails an event that occurs with equal probability for all the units TheprobabilitythatthesystemwillstopduringagivencycleispAnewcyclestartsafter the preceding one has been completed or if the preceding cycle has notbeencompletedafterthedamagedunithasbeenrepairedFindtheexpectationofthenumberofunitssubjecttorepairsatleastonceduringmcycles

2116TherearenunitsoperatingindependentlyofeachotherandcarryingoutaseriesofconsecutivecyclesTheprobabilityoffailureforanyunitduringone cycle is p A new cycle starts after the preceding one is completed(separatelyforeachunit)orafterrepairsiftheprecedingcycleisnotcompletedFindtheprobabilityofthenumberofunitssubjecttorepairsatleastonceifeachunitoperatesformcycles

2117 In an electronic device the number of elements failing to operateduringsometimeintervalobeysPoissonrsquoslawwithparameteraThedurationtmofrepairsdependsonthenumbermofdamagedelementsandisgivenbytm=T(1ndashendashαm)Findtheexpectationofthedurationofrepairsandthelosscausedby delay if the loss is proportional to the square of the duration of repairs

2118AsystemhasnunitsoperatingindependentlyIfatleastoneunitfails

the systemwill stop The probability of occurrence of this event isp and thefailuresofallunitsareequallyprobableAnewcyclestartsafterthecompletionoftheprecedingoneorafterthedamagedunithasbeenrepairediftheprecedingcyclehasnotbeencompleted

Thesystemmustrun2mcyclesandmoreoverafterthefirstmcycles(mltn2)alltheunitssubjecttorepairsatleastoncearediscardedandanumbermofcyclesarerepeatedwiththeremainingunitsunderthepreviousconditionsFindtheexpectationofthenumberofunitsrepairedatleastonceaftertwoseriesofmcycleseach

2119AmarksmanfirestwoseriesofmshotseachatntargetsTheshotsarefiredsuccessivelyateach targetand thedetailedresultsofeachseriesofshots

arenotrecordedThebulletcanstrikewithprobabilityponlythetargetaimedatby themarksmanA target is consideredhit if at leastonebullet reaches itThe secondseries is firedafter the targetshit in the first seriesarenotedTherulesare thesameas in the first seriesexcept that shotsarenot firedat thosetargets hit in the first series Find the expectation of the number of targets hitduringthewholeexperimentforn=m=8andnge2m

2120TwopointsareselectedatrandomonadjacentsidesofarectanglewithsidesaandbFindtheexpectationofthedistancebetweenthesetwopoints

2121Find the expectation of the distance between two randomly selectedpointsonoppositesidesofarectanglewithsidesab

2122Obtaintheformulasfortheexpectationandvarianceofthenumberofoccurrences of an event in n independent trials if the probability for itsrealizationvariesfromonetrialtoanotherandequalspk(k=12hellipn)atthekthtrial

2123 Tenweights are placed on a scale The precision ofmanufacture ofeach weight is characterized by a mean error of 01 g The precision in theprocessofweighing ischaracterizedbyameanerrorof002gFind themeanerrorinthedeterminationofthemassofabody

2124 Two points are taken at random on a segment of length l Find theexpectationandvarianceofthedistancebetweenthem

2125 The probability density of a system of random variables (X Y) isspecifiedbytheformula

FindtheexpectationandvarianceoftherandomvariableZ=aX+bY2126ArandomvariableXobeysthenormaldistributionlaw

EvaluatetheexpectationandvarianceoftherandomvariableY=|X|2127A randomvariableX obeys Poissonrsquos law Find the expectation and

varianceoftherandomvariableY=cosbX2128Thedistancefromalighthouseisgivenasthearithmeticmeanofthree

measurementsTherelationbetweenerrorsdependsontherateofmeasurementsandischaracterizedbythefollowingvaluesofthecorrelationcoefficients

(a)forarateof3secr12=r23=09r13=07(b)forarateof5secr12=r23=07r13=04(c)forarateof12secrij=0jnei

Determinethevalueofthevarianceforthearithmeticmeaninmeasurementswith different rates if the errors of each measurement are characterized by avarianceof30sqm

2129 A random variable X obeys a distribution law with a probabilitydensity

TheprobabilitydensityofarandomvariableYisgivenbytheformula

DeterminetheexpectationandvarianceoftherandomvariableZ=XndashY iftherandomvariablesXandYareindependent

2130Givenarandompointintheplanewithcoordinates(XY)and =0=ndash10σx=100σy=20kxy=0 find theexpectationandvarianceof the

distanceZ from theorigin to theprojectionof this point onOZwhichmakeswithOXanangleα=30deg

2131DeterminethecorrelationcoefficientfortherandomvariablesXandYifXisacentralizedrandomvariableandY=Xnwherenisapositiveinteger

2132FindtheexpectationandvarianceofarandomvariableZ=X(Yndash )iftheprobabilitydensityofthesystem(XY)isgivenbytheformula

2133Awheel is spunand thenslowsdownbecauseof frictionWhen thewheel stops a fixed radius a makes a random angle φ with the horizontal

diameter φ is distributed uniformly over the interval 0 to 360deg Find theexpectation and variance of the distance from the end of radius a to thehorizontaldiameter

2134 As a result of a central force a mass point describes an elliptictrajectoryThemajorsemi-axisaandtheeccentricityoftheellipseeareknownAssumingthatitisequallyprobabletosightthemovingpointatanyinstantfindtheexpectationandvarianceof thedistanceat the instantofobservation if theobserverislocatedatthecenterofattractionatoneofthefocioftheellipseandthedistanceR to the point is givenby the formulaR =α(1ndashe2)(1 ndash cos u)whereuistheanglemadebytheradius-vectorRwiththemajoraxisa(InthecaseofamotioninacentralfieldthesectorvelocityR2dudt=const)

22 THEDISTRIBUTIONLAWSOFFUNCTIONSOFRANDOMVARIABLES

BasicFormulasThe probability density fy(y) of a random variableY whereY =φ(X) is a

monotonic function (ie the inverse function X = ψ(Y) is single-valued) isdefinedbytheformula

If the inverseX =ψ(Y) is not single-valued ie to one value of Y therecorrespondseveralvaluesofXmdashψ1(y)ψ2(y)ψ3(y)hellipψk(y)(Figure21)mdashthentheprobabilitydensityofYisgivenbytheformula

For a function of several random arguments it is proper to start from theformulafor thedistributionfunctionFy(y)ForexampleY=φ(X1X2)and letfx(x1x2)betheprobabilitydensityofthesystemofrandomvariables(X1X2)IfDyisaregionintheplaneX1OX2forwhichYltythenthedistributionfunctionis

FIGURE21

andtheprobabilitydensityoftherandomvariableYisfy(y)=dFy(y)dy In thegeneral case if the Jacobiandeterminant for the transformationof the randomvariables(X1X2hellipXn)totherandomvariables(Y1Y2hellipYn)is

andifthisisaone-to-onetransformationthen

inwhichx1hellipxnareexpressedintermsofy1hellipyn


Example 221 A straight line is drawn at random through a point (0 l)(Figure22)Findtheprobabilitydensityoftherandomvariableη=lcosφ

SOLUTIONTheangleφ isarandomvariableuniformlydistributedovertheinterval(0π)(Figure22)

Sinceheretheinverseψ(η)issingle-valued(whenangleφvariesfrom0toπthefunctiondecreasesmonotonically)todeterminetheprobabilitydensityforηweapplytheformula

where

FIGURE22

Finallywehave

SimilarlyonecansolveProblems222225to227229to2213and2219

Example222ArandomvariableYisgivenbytheformula

Find the probability density of Y if X is a normal random variable withparameters =0D[X]=1

SOLUTION In this example the inverse is two-valued (Figure23) since toonevalueofYtherecorrespondtwovaluesofX

and

bythegeneralformulawehave


Example 223 The position of a random point with coordinates (XY) isequallyprobableinsideasquarewithside1andwhosecentercoincideswiththeoriginDeterminetheprobabilitydensityoftherandomvariableZ=XY

SOLUTIONWeshallconsiderseparatelytwocases(a)0ltzlt14and(b)ndash14ltzlt0Forthesecasesweshallconstructintheplanetwohyperbolaswithequationsz=xy

FIGURE23

InFigure24AandBaregionisshadedinsidewhichtheconditionZltzissatisfied

ThedistributionfunctionoftherandomvariableZisdefinedfor0ltzlt14as

where istheareaoftheregionDprimezforndash14ltzlt0

Differentiatingtheseexpressionswithrespecttozweobtaintheprobabilitydensity

for0ltzlt14

forndash14ltzlt0

FIGURE24A

FIGURE24B

FinallytheprobabilitydensityfortherandomvariableZ=XYcanbewrittenasfollows

Problems2216to2219and2221aresolvedsimilarly

Example224Asystemof randomvariables (XY) isnormallydistributedwiththeprobabilitydensity

Findtheprobabilitydensityofthesystem(RΦ)if

SOLUTIONTodeterminetheprobabilitydensityof thesystem(RΦ)applytheformula

where istheJacobiandeterminantofthetransformationfromthegivensystemtothesystem(RΦ)

Therefore

TherandomvariablesRandΦareindependentsothat

wherefr(r)=(rσ2)endashr22σ2isRayleighrsquoslawandfφ(φ)istheuniformdistribution

lawSimilarlyonecansolveProblems22222223and2225to2227

PROBLEMS

221 The distribution function of a random variable X is Fx(x) Find thedistributionfunctionoftherandomvariableY=aX+b

222Giventheprobabilitydensityf(x)ofarandomvariableX(0ltxltinfin)findtheprobabilitydensityoftherandomvariableY=lnX

223Find theprobabilitydensityof the randomvariableZ =aX2 ifX is anormalrandomvariable =0D[X]=σ2andagt0

224EvaluatetheprobabilitydensityoftherandomvariableY=|X|ifXisanormalrandomvariableforwhich =0andthemeandeviationEisgiven

225ArandomvariableXisuniformlydistributedovertheinterval(01)andrelatedtoYbytheequationtantanπY2=eXFindtheprobabilitydensityoftherandomvariableY

226FindtheprobabilitydensityofthevolumeofacubewhoseedgeXisarandomvariableuniformlydistributedintheinterval(0a)

227A straight line is drawn at random through the point (0 l) Find theprobabilitydensityofthex-interceptofthislinewiththeOx-axis

228A randomvariableX is uniformly distributed over the interval (ndashT2T2)FindtheprobabilitydensityoftherandomvariableY=asin(2πT)X

229ArandomvariableXobeysCauchyrsquosdistributionlaw

FindtheprobabilitydensityoftherandomvariableYif(a)Y=1ndashX3(b)Y=aX2(c)Y=arctanX

2210 Determine the probability density of the random variable Y = XnwherenisapositiveintegeriftheprobabilitydensityforXis

2211A randomvariableX is distributed over the interval (0infin)with theprobability density fx(x) = endashx Evaluate the probability density of the randomvariableYif(a)Y2=XandthesignsofYareequallyprobable(b)Y=+

2212ArandomvariableXobeysPearsonrsquosdistributionlaw

FindtheprobabilitydensityoftherandomvariableY=arcsinX2213 A random variableX is uniformly distributed in the interval (0 1)

EvaluatetheprobabilitydensityoftherandomvariableYif

2214 The random variables X and Y are connected by the functionaldependenceY=Fx(X)TherandomvariableX isuniformlydistributedovertheinterval(ab)andFx(x)isitsdistributionfunctionFindtheprobabilitydensityofrandomvariableY

2215ArandomvariableX isuniformlydistributedovertheinterval(01)Assume that there is a function ft(t) ge 0 satisfying the condition

TherandomvariablesXandYarerelatedbytheequation

Prove that ft(t) is the probability density of randomvariableY

2216Asystemofrandomvariables(XY)obeysthenormaldistributionlaw

WhatdistributionlawdoestherandomvariableZ=XndashYobey2217FindtheprobabilitydensityoftherandomvariableZ=XYif(a)theprobabilitydensityf(xy)ofthesystemofrandomvariables(XY)is

given(b)XandYareindependentrandomvariableswithprobabilitydensities

(c)XandY are independentnormal randomvariableswith = =0and

variances and respectively(d)XandYareindependentrandomvariableswithprobabilitydensities

2218FindtheprobabilitydensityoftherandomvariableZ=XYif(a)theprobabilitydensityf(xy)ofthesystemofrandomvariables(XY)is

given(b) X and Y are independent random variables obeying Rayleighrsquos

distributionlaw

(c)XandYareindependentrandomvariableswithprobabilitydensities

(d)thesystemofrandomvariables(XY)obeysthenormaldistributionlaw

2219 Find the probability density for the modulus of the radius-vectorif

(a)theprobabilitydensityf(xy)forthesystemofrandomvariables(XY)isgiven

(b)therandomvariablesXandYareindependentandobeythesamenormaldistributionlawwithzeroexpectationandmeandeviationE

(c)theprobabilitydensityforthesystemofrandomvariables(XY)isgivenbytheformula

(d)X andY are independent normal randomvariableswith the probabilitydensity

(e) the random variables X and Y are independent and obey a normaldistributionlawwith = =0andvariances and respectively

2220Asystemofrandomvariables(XY)hastheprobabilitydensity

Find the linear transformation leading from random variables X Y to theindependentrandomvariablesUVEvaluatethemean-squaredeviationsofthenewrandomvariables

2221Bothrootsofthequadraticequationx2+αx+β=0cantakeallvaluesfromndash1to+1withequalprobabilitiesEvaluatetheprobabilitydensityforthecoefficientsαandβ

2222The rectangular coordinates (XY) of a random point are dependentrandom variables and are given Find the probabilitydensityofthepolarcoordinates(Tφ)ofthispointif

WhatdistributionlawsdoTandΦobeyifrxy=02223 LetS = S0 +V0t + (At22) whereS0V0 andA are normal random

variables whose expectations and covariance matrix are known Evaluate theprobabilitydensityf(s|t)

2224 Find the probability density of the nonnegative square root of thearithmetic mean for squares of normal centralized random variables

ifthevarianceD[Xj]=σ2(j=12hellipn)2225The rectangular coordinatesof a randompoint (X1X2hellipXn)have

theprobabilitydensity

Find theprobabilitydensity forn-dimensional sphericalcoordinatesof thispointRΦ1Φ2hellipΦnif

2226Twosystemsofrandomvariables(X1X2hellipXn)and(Y1Y2hellipYn)arerelatedbylinearequations

where|aij|ne0Evaluatetheprobabilitydensityfy(y1y2hellipyn)iftheprobabilitydensityfx(x1x2hellipxn)isgiven

2227 Find the distribution law of the system of random variables (R Θ)where istheradius-vectorofarandompointinspace and Θ = arcsin YR is the latitude if the probability density of therectangularcoordinates(XYZ)isf(xyz)

23 THECHARACTERISTICFUNCTIONSOFSYSTEMSANDFUNCTIONSOFRANDOMVARIABLES

BasicFormulasWedefine the characteristic function of a systemof randomvariables (X1

X2hellipXn)astheexpectationofthefunctionexp whereuk(k=12hellipn)arerealquantitiesand

Forcontinuousrandomvariables

The characteristic function of a system of independent random variablesequals the product of the characteristic functions of the random variablescontainedinthesystem

For a multidimensional normal distribution with expectationsandcovariancematrix

wehave

Iftheappropriatemomentsofasystemofrandomvariablesexist

IftherandomvariableY=φ(X)then

Thecharacteristicfunctionofasystemofrandomvariables(Y1Y2hellipYn)ofwhicheachisafunctionofotherrandomvariables

equals

The characteristic function of a subsystem of random variables can beobtained from the characteristic functions of the system by replacing thevariablesukcorrespondingtorandomvariablesnotinthesubsystembyzeros


Example 231 A particle starts from the origin and moves in a certaindirection foradistance l1Then it changes itsdirectionmany timesmakingarandomwalkforadistancel2thenforadistancel3andsoforthThetrajectoryof thewanderingparticle consists thusof segmentsof lengths l1 l2hellip ln thedirectionofeachbeingdeterminedbytheangleαkmadewiththeOx-axisTheseanglesareuniformlydistributedintheinterval(02π)andtheyareindependentFind the characteristic function of the coordinate X of the endpoint of thetrajectoryandthecorrespondingprobabilitydensity

SOLUTIONThecoordinateX isdeterminedas thesumof theprojectionsofsegmentslkontheOx-axis

Sinceαkareindependent

and

Therefore

whereJ0istheBesselfunctionofthefirstkindofzeroorderFromthis

or

Example232Given thecovariancematrix ||krs||ofasystemofsixnormalrandomvariablesX1X2hellipX6withzeroexpectationsevaluatetheexpectation

oftheproduct byapplyingthemethodofcharacteristicfunctions

SOLUTIONTheexpectationM[ ] isdeterminedbythedistributionofthesubsystem(X2X3X4)Thecharacteristic functioncorresponding to thissubsystemhastheform

Therequiredexpectationcanbeobtainedbydifferentiatingthecharacteristicfunctionfourtimes

Thefirstmethod Ifweexpandthecharacteristicfunction inapowerseriesaccording to its exponent then we find that in calculating the desired mixedpartialderivativeforu2=u3=u4=0onlyonetermoftheexpansionisdifferentfromzero

Themixedderivativeof thesquareof thepolynomial foru2=u3=u4 = 0will have terms different from zero if before differentiation they wereproportionalto thatis

ThesecondmethodForconvenienceweintroducethenotation

Then

whichimpliesthat


PROBLEMS

231Provethatthecharacteristicfunctionofthesumofindependentrandomvariablesistheproductofthecharacteristicfunctionsofitsterms

232Given thecharacteristicfunctionofthesystem(X1X2hellipXn)findthecharacteristicfunctionofthesumZ=X1+X2+middotmiddotmiddot+Xn

233 Find the characteristic function of the linear function of the random variables X1 X2 hellip Xn whose

characteristicfunctionsaregiven234 Find the characteristic function for the square of the deviation of a

normal random variable from its expectation and themomentsofY

235FindthecharacteristicfunctionoftherandomvariableY=aF(X)+bwhereXisarandomvariableandF(x)isitsdistributionfunction

236 Find the characteristic function of the random variable Y = ln F(X)whereX is a random variable andF(x) its distribution function Evaluate themomentsofY

237FindthecharacteristicfunctionoftheprojectionofasegmentaontheOy-axis if the angle made by this segment with the Oy-axis is uniformlydistributed in the interval (0 2π) Evaluate the probability density of theprojection

238 Find the characteristic function of a system of two random variablesobeyingthenormaldistributionlaw

239Findthecharacteristicfunctionofasystemofnrandomvariables(X1X2hellipXn)obeyinganormaldistributionlawiftheexpectationsoftherandomvariablesformingthesystemareallequaltoaandtheircovariancematrixis

2310Findthecharacteristicfunctionof

inwhich(X1X2hellipXn)isasystemofnormalrandomvariblesand

2311 Using the method of characteristic functions findifX1X2arenormalrandomvariablesforwhich

2312 Applying the method of characteristic functions evaluate (a)

(b) if X1 X2hellip X3 are normal random variables for which

andk12k13k23arethecovariances

betweenthecorrespondingrandomvariables2313ApplyingthemethodofcharacteristicfunctionsevaluateM[X1X2X3]

ifX1X2X3arenormalcentralizedrandomvariables2314UsingthemethodofcharacteristicfunctionsexpressM[X1X2X3X4]

in termsof theelementsof thecovariancematrixkmlof thesystemof randomvariables(X1X2X3X4)whoseexpectationsarezero

2315Provethatthecentralmomentofevenorderofasystemofnnormalrandomvariablesisgivenbytheformula

wherer1+r2+ middotmiddotmiddot+rn=2s and the summation is extendedover all possiblepermutationsof2sindicesm1m2hellipmnand l1 l2hellip lnofwhichr1 indicesequal1r2indicesequal2helliprnindicesequaln

2316Givenasystemofdependentnormalrandomvariables(X1X2hellipXn)

prove that the randomvariable also obeys a normaldistributionlaw

2317Theoutputofafactoryconsistsofidenticalunitseachofwhichintherthquarteroftheyear(r=1234) iswithprobabilityproffirstqualityandwithprobabilityqr=1ndashprofsecondqualityAn itemof firstqualitycostsS1dollars and an item of second quality S2 dollars Evaluate the characteristicfunctionofthesystemofrandomvariables(XY)whereX is thecostof itemsproducedduringthefirstthreequartersoftheyearandYthecostduringthelastthreequartersof theyearEvaluate the covarianceofXandYThenumberofitemsproducedintherthquarterisNr

24 CONVOLUTIONOFDISTRIBUTIONLAWS

BasicFormulasThe operation of finding the distribution law of a sum of mutually

independentrandomvariablesintermsofthedistributionlawsofitssummandsis called convolution (composition) of distribution laws If X and Y are

independent discrete random variables the distribution series of the randomvariableZ=X+Yisgivenbytheformula

where the summation is extended over all possible values of the randomvariables

IfXandY arecontinuous randomvariables theprobabilitydensity for therandomvariableZ=X+Yis

andthedistributionfunctionFz(z)isdeterminedbytheformula

Theprobabilitydensity fy(y)ofasumof independent randomvariablesX1X2hellipXn (Y=X1+X2+ middotmiddotmiddot+Xn) canbe foundeitherbyusing thecharacteristicfunctionsinaccordancewiththeformula

where

or by successive applications of the convolution formula for two randomvariables


Example241 Find the probability density of the sumof two independentrandomvariablesZ=X+YwhereX isuniformlydistributedover the interval(01)andYhasSimpsonrsquosdistribution(Figure25)

FIGURE25

SOLUTIONSincethefunctionsfx(x)andfy(y)aredifferentfromzeroonlyforparticular values of their arguments it is more convenient first to find thedistributionfunctionoftherandomvariableZWehave

whereDzistheregioninsidewhichx+yltzandnoneofthefunctionsfx(x)andfy(y)vanishes(Figure26)

Theshapeoftheintegrationdomaindependsonwhichofthethreeintervals(0 1) (1 2) or (2 3) contains zComputing the integrals for these casesweobtain

Bydifferentiationwithrespecttozwefindtheprobabilitydensity

FIGURE26

FIGURE27

Thefunctionsfx(x)fy(y)andfz(z)arerepresentedinFigure27Problems241242244and248canbesolvedsimilarly

Example242ApointC ischosenatrandomonasegmentA1A2of length2L The possible deviation of the midpoint of segment F1F2 = 2B from themidpoint ofA1A2 has a normal distribution with mean deviation E Find theprobability that thedistancefromC to themidpointof segmentF1F2 does notexceedagivenquantity(d+B)

SOLUTION Let X denote the random deviation of the point C from themidpointofA1A2 and letY be thedeviationof themidpoint ofF1F2 from the

midpoint of A1A2 (Figure 28) Then the deviation of the point C from themidpointofsegmentF1F2isZ=YndashXSincethefunctionfy(y)doesnotvanishontherealaxis

ThedistancefromCtothemidpointofF1F2willnotexceedthequantityd+Bif|z|ltd+BThereforetheprobabilityofthiseventisgivenbytheformula

FIGURE28

In a similar manner Problems 243 245 to 247 2413 to 2415 can besolved

Example243Twogroupsof identical itemsofn1andn2 items each aremixed together The number of defective items in each group (X and Yrespectively)hasthebinomialdistribution

FindthedistributionseriesoftherandomvariableZ=X+YSOLUTIONFortheprobabilityP(Z=z)tobedifferentfromzeroZmustbe

integral-valuedandlieontheinterval(0n1+n2)Applyingthegeneralformulaandtakingintoaccountthat0lexlezweobtain

(Theequality canbeprovedforexamplebyinductionFirstoneprovesitforn1=1andforanyn2)

This problem can also be solved by using characteristic functions For therandomvariablesXandYwehave

SinceXandYarebyhypothesisindependentwehave

From this it follows that the random variable Z also has a binomialdistribution

SimilarlyonecansolveProblems2412and2416to2421

Example244LetX1X2hellipXnbe independent randomvariableseachofwhichobeysPoissonrsquoslaw

withthesameparameteraFind the distribution series of the random variable and

provethatthecentralizedandnormalizedrandomvariable fornrarrinfinhasanormaldistribution

SOLUTIONWefindthecharacteristicfunctionoftherandomvariableXj

SincetherandomvariablesXjareindependent thecharacteristicfunctionofYisgivenbytheformula

Consequently the random variable Y has Poissonrsquos distribution law withparameternaUsethenotation TherandomvariableZ isobtainedasaresultofnormalizingandcentralizingtherandomvariableYItisknownthatforPoissonrsquoslawtheexpectationandvariancearenumericallyequalquantitiesbothequaltotheparameterofthislawThus

EvaluatethecharacteristicfunctionofZ

Consequently

ThelimitofEz(t)isthecharacteristicfunctionoftherandomvariablewithanormaldistributionwithexpectationzeroandvarianceone


PROBLEMS

241Find theprobability density of the sumof two independent variableseachofwhichisuniformlydistributedovertheinterval(ab)

242Findtheconvolutionoftwouniformdistributionswithparametersaandb (b gt a) if the dispersion centers for both distributions coincide and theparameterofauniformdistributionlawisdefinedasbeinghalfthelengthoftheintervalofthepossiblevaluesofarandomvariable

243TherandomvariableXobeysanormaldistributionlawwithparametersandσxYobeysauniformdistributionlawwithparameter(bndasha)2and =(a

+b)2FindtheprobabilitydensityoftherandomvariableZ=XndashYifXandYareindependent

244 Find the probability density of the sum of three independent randomvariableseachofwhichisuniformlydistributedovertheinterval(ab)

245 Find the convolution of a normal law (with expectation andmeandeviation E) and a uniform distribution law given in the interval

Findtherelativeerrorcausedbyreplacingtheresultinglawby a normal law with the same variance and expectation (Perform thecomputationsfor =0l=El=2El=3Eandl=4Eatpointz=0)

246 Find the probability density of the random variableZ =X +Y if therandomvariablesXandYareindependentandobeyCauchyrsquoslaw

247FindtheprobabilitydensityofthesumoftworandomvariablesXandYobeyingthehyperbolicsecantlaw

248LetXandYbeindependentrandomvariableswithprobabilitydensitiesgivenbytheformulas

FindtheprobabilitydensityoftherandomvariableZ=X+Y249Find theprobabilitydensityof thedistancebetween thepointsA1(X1

Y1) and A2(X2 Y2) if the systems (X1 Y1) and (X2 Y2) are independent anduniformlydistributedTheunitdispersionellipsesofthepointsA1andA2havemajor semiaxes (a1b1) and (a2 b2) The anglemade by a1 anda2 is α Thecentersoftheunitellipsescoincide

2410 LetXj(j = 1 2hellip n) be normally distributed independent randomvariables with and D[Xj] = 1 Prove that for the random variable

theprobabilitydensityisdeterminedbytheformula

2411Aninstrumentgivesasystematicerroraandarandomerrorobeyinganormal distribution law with mean deviation E Prove that for E ge d theprobabilityp(a)ofanerrorwithinagiventolerancerangeplusmnd isapproximatelygivenbytheformula

where

2412TwopersonsfireindependentshotseachathistargetuntilthefirsthitisscoredFindtheexpectationandvarianceforthetotalnumberoffailuresandthedistributionfunctionforthenumberoffailuresiftheprobabilityofhittingatargetateachshotisp1forthefirstmarksmanandp2forthesecond

2413 What should be the reserve shear strength of a sample so that theprobability that itwill support a load is at least98per centTheerrors in thedetermination of the given load and of the maximal load obey a normaldistributionwithmean deviations whereand aretheexpectationsforthegivenandmaximalloadsand =20kg2414 A navigational transmitter is installed on each shore of a sound of

width L The transmitters serve the ships passing through the sound Themaximal ranges of each of the transmitters are independent random variableswithexpectation andmeandeviationEAssumingthatanydistancebetweenthecourseofashipandtheshoresisequallyprobableandthat2 ltLfind(a)theprobabilitythatashipwillbeservedbytwotransmitters(b)theprobabilitythatashipwillbeservedbyatleastonetransmitter

2415 Observer A moves from infinity toward observer B The maximaldistances for sighting each other are independent random variables withexpectations and respectively and mean deviations EA EB Find theprobabilitythatAwillsightBfirst

2416 Find the convolution ofm exponential distributions with the sameparameterλ

2417 Let X and Y be independent random variables assuming integralnonnegativevaluesiandjwithprobabilitiesP(X=i)=(1ndasha)aiandP(Y=j)=(1ndashb)bjwhereaandbarepositiveintegerslessthanoneFindthedistributionfunctionoftherandomvariableZ=X+Y

2418 Let X and Y be independent random variables X assumes threepossible values 0 1 3 with probabilities 12 38 18 and Y assumes twopossiblevalues0and1withprobabilities1323FindthedistributionseriesoftherandomvariableZ=X+Y

2419 Let X Y be independent random variables each of which obeysPoissonrsquosdistribution

FindthedistributionseriesoftherandomvariableZ=X+Y2420LetXj(j=12hellipn)beindependentrandomvariableseachofwhich

takesonlytwovalues1withprobabilitypandzerowithprobabilityq=1ndashpFindthedistributionseriesoftherandomvariable

2421 Let X and Y be independent discrete random variables assumingpositive integral values k from 1 to infin with probability (l2)k Find thedistributionfunctionoftherandomvariableZ=X+Y

25 THELINEARIZATIONOFFUNCTIONSOFRANDOMVARIABLES

BasicFormulasAnycontinuousdifferentiablefunctionwhosederivativeisfiniteatagiven

point and for sufficiently small variations about the point can be replacedapproximatelybya linearfunctionbyusingaTaylorseriesandretainingonlythelineartermsIftheprobabilityissmallthattheargumentsofthefunctionwillassumevalues outside the regionwhere the function canbe considered linearthis functioncanbeexpanded in thevicinityof thepointcorresponding to theexpectations of its arguments The approximate values of the expectation andvarianceinthiscasearegivenby

(a)forthefunctionofonerandomargumentY=φ(X)

(b)forafunctionofmanyargumentsY=φ(X1X2hellipXn)

wherekijdenotesthecovariancefortherandomvariablesXiandXjandare the derivatives computed for values of the arguments equal to theexpectations

Iftherandomargumentsaremutuallyuncorrelatedthen

For more accuracy in the results of linearization in the expansion of thefunctiononemustretainbesidethefirsttwotermssomehigher-ordertermsaswell Ifone retains the first three termsof the series then theexpectationandvariancearedeterminedbytheformulas

(a)forafunctionofoneargumentY=φ(X)

(b) for a function of several random argumentsY =φ(X1X2hellipXn) theexpectationisgivenbytheformula

inthegeneralcaseandbytheformula

inthecasewhentherandomargumentsaremutuallyuncorrelatedIftherandomargumentsaremutuallyindependentthenthevarianceisgivenbytheformula


Example251Theexpectationof thenumberofdefectivedevices isgivenbytheformula

wherePistheprobabilitythatthetrialofonedeviceisconsideredsuccessfulΩis theaveragenumberofsuccessful trialsuntil thefirstfailureoccursN is thenumberofdevicestestedandm is thenumberof trials(successesandfailures)foreachdevice

Usingthe linearizationmethodfind thedependenceof theexpectationandvarianceoftherandomvariableTonmifNPandΩare independent randomvariableswhoseexpectationsandvariancesare

SOLUTION Applying the general formulas of the linearization method weobtain

where

Theapproximatevaluesof theexpectations andvarianceofT fordifferentvaluesofmaregiveninTable8

TABLE8

Similarlyonecan solveProblems251 to251125142517 and2519 to2522

Example252Themaximalaltitudeofasatelliteisgivenbytheformula

where

y0isthealtitudeoftheactivepartofthetrajectorygtheaccelerationofgravityonthesurfaceoftheearthandRtheradiusoftheearth

ThefunctionYcanbelinearizedinthedomainofpracticallypossiblvaluesof the random arguments The initial velocityV and the launchin angleΘ arenormalrandomvariableswithprobabilitydensity

Find theapproximatevalueof thevariance for themaximal altitudeof thesatellite

SOLUTION Since the given function is linearizable in the domain of thepracticallypossiblevaluesoftherandomarguments

wherekvθ=rσvσθ

andλandlarecomputedfor OnecansolveProblems2513and2523inasimilarway

Example253LetXandYbeindependentrandomvariableswithprobabilitydensity

Using the linearization method find the expectation and variance of therandomvariableZ=arctanXYCorrect the resultsobtainedbyusing the firstthreetermsoftheTaylorseries

SOLUTIONUsingthegeneralformulasoflinearizationwehave

where

Thusthelinearizationmethodgives

ConsideringthenexttermofTaylorrsquosseriesweobtain

where

ThereforetakingintoaccountthequadratictermsoftheTaylorseriesweobtain


PROBLEMS

251 The amount of heat Q in calories produced in a conductor withresistanceRbyacurrentIintimeTisgivenbytheformula

The errors in the measurements of I R and T are independent randomvariables with expectations = 10 amps = 30 Ω t = 10 min and meandeviationsEI=01ampER=02ΩET=05secFindtheapproximatevalueofthemeandeviationoftherandomvariableQ

252Thefundamentalfrequencyofastringisgivenbytheformula

wherePisthetensionMthemassofthestringandLthelengthofthestringGiven the expectations and mean-square deviations σp σm σl

find thevarianceof the fundamental frequencycausedby thevariancesof thetension mass and length of the string if the corresponding correlationcoefficientsarerplrpmrml

253Theresistanceofasectionofanelectriccircuitisgivenbytheformula

whereRdenotes theohmicresistanceL the inductanceof theconductorC itscapacityandΩthefrequencyofthecurrent

Evaluate the mean error in the magnitude of the resistance as a result of

errorsinindependentmeasurementsofRLCandΩifoneknowsandthemeandeviationsERELECEΩ

254Iftheelementsofacircuitareconnectedinparalleltheintensityofthecurrentinthecircuitisgivenbytheformula

whereEistheelectromotiveforceacrossthesystemWisitsinternalresistancenisthenumberofelementsandRistheresistanceofanexternalsectionofthecircuit

Using the linearization method find the expectation and variance of theintensityofthecurrentiftherandomvariablesERandWareindependentand

aregiven255ApplyingthelinearizationmethodfindthemeandeviationsExandEy

which characterize the variance of coordinates of a mass point moving in avacuumif

whereVistheinitialvelocityofthepoint( =800msecEv=01percentof)Tisthetimeoftheflight( =40secET=01sec)Θisthelaunchingangle

( =45degE0=4rsquo)andgistheaccelerationofgravityTherandomvariablesVTandθareindependentandnormal256Findtheapproximatevalueofthemeanvalueoftheerrorinestimating

theprojectionV1ofthevelocityofashiponagivendirectionErrorsareduetomeasuringthevelocityVandtheangleqofthecourseHereV1=ndashVcosqEv=1msecEq=1degand themostprobablevaluesofVandqare10msecand60degrespectively(Vandqareindependentnormalrandomvariables)

257 Is the linearizationmethod applicable under the assumptionsmade intheprecedingproblemiftheerrorinthecomputationformulasmustnotexceed02msec

258 Find the approximate value of the mean-square deviations forrectangularcoordinatesofarandompoint

if the randomvariablesHε andβ are independent and their expectations andmean-squaredeviationsareequal respectively to =6200m =45deg =30degσH=25mσβ=σε=0001radians

259 The passage from spherical to Cartesian coordinates is given by theformulas

Theerrors in thedeterminationofΘRandΦare independentwithmean-squaredeviationsσR=10mσΘ=σΦ=0001radiansFindapproximatevaluesfor mean-square deviations of the rectangular coordinates if

2510Theapproximateexpressionforthevelocityofarocketattheendof

theoperationofitsengineisgivenbyTsiolkovskiyrsquosformula

whereUistheeffectivevelocityofgasflowqtheweightoftherocketwithoutfuelandΩtheweightofthefuel

ThevarianceoftheweightofthefuelischaracterizedbythedeviationEΩFindtheapproximatevalueofthemeandeviationofthevelocitycausedbythevarianceoftheweightofthefueliftheexpectation

2511ThealtitudeofamountainpeakHexpressedintermsofthedistanceDontheslopeandtheinclinationangleεis

FindtheapproximatevalueofthemeanerrorinestimatingthealtitudeifED=80mEε=0001degandthemostprobablevaluesare =12300mand =31deg2respectively(TherandomvariablesDandεareindependentandnormal)

2512LetZ=sinXYwhereXandYareindependentrandomvariablesFindtheapproximatevalueofσzif σx=σy=0001

2513ThealtitudeofamountainpeakisgivenbytheformulaH=DsinεTheprobabilitydensityof theerrors inestimating thedistanceD on the slopeandtheinclinationangleεisgivenby

whereσd=40mσε=0001radians =10000mand =30degFindtheapproximatevalue for themeandeviationof theerrorsmade inestimating thealtitude

2514The distanceD1 (Figure29) is determined by a radar stationwhoseerrorshavethemeandeviationEp=20mDistanceD2canbedeterminedeitherwitharangefinderwhichgiveserrorswithmeandeviationED=40morbytheformula

FIGURE29

FindwhichmethodofdeterminationofdistanceK2CismoreaccurateiftheerrorsinestimatingthedistancebetweenK1andK2havemeandeviationEd=50m

2515Retaining the first three terms of the expansion of the functionY =φ(X)inaTaylorseriesfindtheexpectationandvarianceoftherandomvariableYifXobeysanormaldistributionlaw

2516Theareaofatriangleisgivenbytheformula

RetainingallthetermsoftheTaylorseriesofthefunctionS=φ(γ)uptoγ3inclusivefindtheexpectationoftheareaofthetriangleandthevarianceofitsareacausedbythedispersionof theangle if therandomvariableγisnormallydistributedand andD[γ]aregiven

2517InthetriangleABC(Figure30)thesideaandtheoppositeangleαarerandom variables which can be considered uncorrelated and normal Find anapproximatevaluefortheexectationXoftheangleanditsmeandeviationifthe

base b is known and the expectations and mean deviations of the randomvariablesaandαareknown

FIGURE30

2518ArandomvariableXobeysthenormaldistributionlaw

Find an approximatevalue for the expectation andvarianceof the randomvariableY=1XRetaintwoandthenthreetermsoftheTaylorseries

2519The radius of a sphere can be considered a normal randomvariablewith expectation and variance Find the expectation andvarianceofthevolumeofrsquothespherebyusingtheexactformulasComparetheresultsobtainedwiththoseofthelinearizationmethod

2520Todetermine thevolumeofaconeonemeasures(a) thediameterofthe base and the height (b) the diameter of the base and the length of thegeneratorInwhichofthesetwocasesistheerrorinthedeterminationofvolumesmalleriftheexpectationfortheheightis =8dmforthediameterofthebased=12dmforthelengthofthegenerator =10dmandσk=σd=σl=01dm

2521Inaweighingprocessoneusesabarwhoseaveragediameteris2mmWhatisthemeanerrorifthemeandeviationofthediameteroftherollis004mmandthedensityofthemetalofwhichtherollismadeis112gccFiftybarsareusedintheprocessofweighing

2522Theaccelerationgofgravityiscomputedbytheformulag=4π2LT2

whereL is thelengthofaphysicalpendulumandT itsperiodFindthemeanerroringifameasurementofthelengthofthependulumwithmeanerrorEL=5mmyieldsL=5mandthemeasuredperiodofoscillationis45secTheperiodofoscillationofthependulumisestimatedforthedurationofn=10completedisplacementsmeasuredwithameanerrorEt=01secandthemeanerrorindetermining the instant when the pendulum passes through a position ofequilibriumisEt=05percentT

2523 Using the linearization method find an approximate value for thevarianceoftherandomvariable ifX=sinVY=cosVtherandomvariableVisuniformlydistributedovertheinterval(0π2)andkisaknownconstant

26 THECONVOLUTIONOFTWO-DIMENSIONALANDTHREE-DIMENSIONALNORMALDISTRIBUTIONLAWSBYUSEOFTHENOTIONOFDEVIATIONVECTORS

BasicFormulasAny two-dimensional (three-dimensional) normal distribution law can be

considered as the convolution of two (three) degenerate normal distributionlawsdescribingthedistributionofindependentobliquecoordinatesofarandompoint in the plane (space) if the coordinate axes are chosen as conjugatedirectionsoftheunitdistributionellipse(ellipsoid)1

Adegeneratenormaldistribution law isuniquelycharacterizedbyavectorpassingthroughthedistributioncenterofthislawinthedirectionofoneoftheconjugatediametersoftheunitellipseandequalinmagnitudetothisdiameterAvectordefinedinthiswayiscalledadeviationvector

Theconvolutionofnormaldistributionsintheplane(space)isequivalenttothe convolution of deviation vectors The convolution of normal distributionslyinginoneplaneandgivenbydeviationvectorsai(i=12hellipk) is formedaccordingtothefollowingrules

(1)thecoordinates ofthecenterofthecompounddistributionaregivenbytheformulas

where arethecoordinatesoftheoriginofthedeviationvectorai(2)theelementskijofthecovariancematrixofthecompounddistributionare

givenbytheformulas

whereaixandaiyaretheprojectionsofthedeviationvectoraiontheaxisofanarbitrarilyselecteduniquerectangularsystemofcoordinates

(3) the principal directions (ξ η) of the compound distribution theircorrespondingvariances andtheangleαmadebytheaxisOξwithOxaredeterminedbytheformulas

FIGURE31

whereαisanyoftherootsoftheequation

Theprincipalsemiaxesoftheunitellipseare

Ifaandbare theprincipalsemiaxesof theunitellipse ifmandnare twoconjugatesemiaxesofthesameellipseifαandβaretheanglesmadebynandmwiththesemi-axisaandifβ+αistheanglebetweentheconjugatesemiaxestheninaccordancewithApolloniusrsquotheorem(Figure31)

where

TheconvolutionofdeviationvectorsinspaceisformedfollowingthesamerulesItisconvenienttoperformthenecessarycomputationsbyusingTable9

TABLE9

Theelementsofthecovariancematrix||ki||ofthecompounddistributionlawaredeterminedbytheformulas

The last two columns of Table 9 serve for checking the accuracy ofcomputationsthefollowingequalitymustbesatisfied

The variances ξ η ζ with respect to the principal directions of thecompounddistributionellipsoid aregivenbytheformulas

whereabc are theprincipal semiaxesof theunit ellipsoidof the compounddistributionandarerelatedtotheroots(u1u2u3)oftheequationu3+pu+q=0asfollows

Therootsofthecubicequationcanbefoundeitherfromspecialtablesortheformulas

where

Thedirectioncosinesofaxesξηζ in thecoordinate systemOxyz are thesolutionsofasystemofthreeequations(i=123)

where

andαij denotes the cosine of the anglemade by the ith coordinate axis of thesystemOξηζwiththejthaxisofthesystemOxyz


Example 261 The position of a point A is defined from a point ofobservationObydistanceOA=Dand theangulardeviationfromareferencelineOB

Themeanerrorinestimatingthedistanceis100kpercentofthedistancethemean error in estimating the angular deviation is ε radiansTheerrormade inrepresentingthepoint^onachartobeysanormaltirculardistributionwithmeandeviationrtheerrorinthepositionofthepointOalsoobeysanormalcirculardistribution law with mean deviation R Find the compound distributioncharacterizing theerror inpositionresultingfromtherepresentationofpointAonthechartHowwilltheprobabilitythatpointAliesinarectangleofsize100times100sqmchangeifDdecreasesfrom20to10km(r=20mR=40mε=0003k=0005)

SOLUTIONIndependentdeviationvectorskDrandRactalongthedirectionofOAandperpendiculartoitthereacttheindependentdeviationvectorsεDrandR2Thedistributionof theerrorsmade in thepositionofAon thechart isdefinedbyaunitellipsewithsemiaxes

andconsequently

FordistanceOA=20000m

Ifthedistancebecomes10000m

Example262ThepositionofapointKintheplaneisdefinedbymeasuringthedistancefromittotwopointsMandNThecoordinatesofthepointobeyanormaldistribution lawgivenbyprincipalsemiaxesa=60mandb=40mandangleα1=47deg52primebetweenthesemi-axisaandthedirectionofNK

HowwillthedistributionofcoordinatesofpointKchangeifthemeanerrorfordistanceMKdecreasestoone-half

SOLUTIONThedeviationerrorsofthecoordinatesofKarisingfromerrorsinthemeasurementsofMKandNKaretheconjugatesemiaxesmandnofaunitellipsedirectedalongthenormalstoMKandNKrespectively(seeFigure31)Thereforea=90degndashα1=42deg8prime

The principal semiaxes of the unit ellipse of the new distribution can bedetermined ifoneconsiders the fact that theconjugate semiaxesof thisellipsearethesegmentsn2=240mandm=538mtheanglebetweenthembeingasbeforeequaltoα+β=68deg18primeUsingApolloniusrsquotheoremhereweobtain

thatis

Example 263 Find the covariance matrix of a three-dimensionaldistributionrepresentingtheconvolutionoffourdegeneratenormaldistributionswiththefollowingdeviationvectors(Table10)

TABLE10

Findtheprincipalsemiaxesoftheunitcompoundellipsoidandthedirectioncosinesoftheanglesbetweenthemajorsemiaxesandtheaxesofcoordinates

SOLUTION (1)Thecomputationof theelementsof thecovariancematrix isgiveninTable11

TABLE11

Check

(2) The computation of the principal semiaxes of the unit compoundellipsoidproceedsasfollows

Accordingtotheprecedingformulaswefind

(3) The computation of the cosines of the angles made by the principalsemiaxesawiththeaxesofcoordinatesproceedsasfollows

Weformthesystemofequations

Fromthefirsttwoequationswefind

andfromthirdequation

Thus

SimilarlyonecansolveProblem269

PROBLEMS

261 Find the convolution of two deviation vectors c1 and c2 if the anglebetween them is γ = 30deg c1 = 30m c2 = 40m and the distribution centerscoincide

262Solvetheprecedingproblemforγ=0degandγ=90deg263 Find the compound distribution that is the convolution of deviation

vectorsai lying in the sameplane if theirmagnitudesareai and the anglesαibetweenaiandthepositivedirectionoftheaxisofabscissasaregiveninTable12

TABLE12

264Find theunit ellipseof the compoundvariance lawof thepoints in aplaneobtainedfromthecompositionofthefollowingdeviationvectorslyinginthisplane(Table13)

TABLE13

265FindtheconvolutionofthedeviationvectorΔ(Δ=18m)makinganangleβ=75degwiththedirectionofOxandanormaldistributiongivenbyaunitellipseoneofthesemiaxesofwhichcoincideswithOxandhaslengtha=30mandtheotherofwhichhaslengthb=20m

266Findtheconvolutionoftwonormaldistributionsintheplaneif(a)theprincipalsemiaxesoftheunitellipsesarea1=b1=50ma2=b2=25m(b)theprincipalsemiaxesoftheunitellipsesarea1=50mb1=25ma2=50mb2=25miftheanglebetweena1anda2is30deg

267 The coordinates of a random point in the plane obey a normaldistributionlawgivenbyaunitellipsewithprincipalsemiaxesa=24mb=7mFindtheprobabilityofhittingadiamondwithside2l=60mandacuteangleγ=34deg3Thecenterofthediamondcoincideswiththecenterofthedistributionandtheadjacentsidesofthediamondareparalleltotwoconjugatesemiaxes

268 Find two deviation vectors equivalent to a normal distribution in theplanecharacterizedbyaunitellipsewithprincipalsemiaxes80mand60mifoneofthedeviationvectorsmakesanangleof30degwiththemajorsemi-axis

269 The coordinates of a ship are determined by a radar station whichestimates thedistance froma referencepointon theshoreand thedirectionofthesightingTheerrorsinmeasurementaregivenbyaunitellipsewithprincipalsemiaxesEx = 80 m in the direction of the axisOx andEz = 30 m in thedirectionofOzTheunitellipseoftheerrorsmadeinestimatingthecoordinatesof the reference point and caused by inaccurate knowledge of its position hasmajorsemiaxesE1=100mE2=40mandE1makesanangleof20degwiththeaxisOx

Find(a)theprobabilitydensityforthecompounderrorsmadeindeterminingthe position of the ship in the system of coordinates xOz (b) the principalsemiaxesandtheorientationwithrespecttotheaxisOxoftheunitellipseofthecompounderrorsinthecoordinatesoftheship

2610Theerrorsindeterminingthepositionofashipatseaareduetothreedeviationerrorswhosemagnitudesanddirectionswithrespect to themeridian

aregiveninTable14

TABLE14

Findtheunitellipseoftheerrorsindeterminingthepositionoftheship2611FindthedistributionlawforthecoordinatesofapointCbysightingit

from two points A andB if the base the angles β1 and β2 and the meanangular errors in sighting from both points Eβ1 = Eβ2 = Eβ are given ThepositionsofAandBareknownwithcertainty(Figure32)

2612Under the assumptionsmade in theprecedingproblem compute themajorsemiaxesoftheunitellipseanditsorientationwithrespecttodirectionABfor =15kmβ1=60degβ2=75degEβ1=Eβ2=00005

2613Under the assumptionsmade in Problems 2611 and 2612 find thecompounddistributionlawfortheerrorsofcoordinatesofpointCwithrespectto A if beside the errors in sighting Eβ1 and Eβ2 there is given as well thedistributionlawfortheerrorsinthepositionofpointBwithrespecttoAwiththemajorsemiaxesalongthebaseE1=30mandperpendiculartothebaseE2=15m

2614Todetermine theactual courseof a shipand itsvelocityonemakestwoestimatesofthepositionoftheship(atthepointsA1andA2)withrespecttosomereferencepointslocatedontheshoreandduringanintervaloftimeτ=20secThedistributionoftheerrorsinthepositionoftheshipiscircularwiththeradius of the unit circle r = 30 m Find the mean error in estimating themagnitude of the velocity and the course of the ship if the distance A1A2 ismeasuredasD=1000m

2615 The coordinates of a ship at time t = 0 are known with an errorobeyinganormalcirculardistributionwiththeradiusoftheunitcircleof100mThemeanerrorinthemagnitudeofthevelocityis2msecrepresenting10percentof itsvelocityandthemeanerror inestimatingitscourse is008radiansCalculatetheunitellipseoftheerrorsmadeinthepositionoftheshipattimet=1min

2616Thepositionofameteorologicalballoonattheinstantofobservation

is known with an error obeying a normal spherical distribution law with theradius of the unit sphere equal to 50m the velocity of the balloon is knownwithmean error 2msec The errors in finding the velocity vector in a planeperpendiculartoitscoursearegivenbyanormaldistributionlawwithradiusofthe unit circle equal to 3 msec Find the unit ellipsoid of the errors in thepositionoftheballoon20secondsafterthecoordinatesandthevelocityvectorhavebeendetermined

FIGURE32

2617Findtheprobabilitydensityforthesumoftworandomnormalvectorsin the spaceOxyz and a random vector in the planeOxz for which the firstmomentsare

respectively and the covariancematrices for the projections of the vectors onaxesofcoordinatesare

Therandomvectorsaremutuallyindependent2618FindtheconvolutionofthedeviationvectorxparalleltotheaxisOx

=25Ex=40ofanormaldistributionintheplanexOywiththeunitellipse

andthenormaldistributioninspacewiththeunitellipsoid

ifxyzaretherectangularcoordinatesofapointinspace2619Constructthecovariancematrixofasystemofthreerandomvariables

(the coordinates of a point in space) that corresponds to the resultant of thefollowingdeviationvectors(Table15)

TABLE15

2620Under the conditionsof theprecedingproblemdetermine themajor

semiaxes of a unit joint distribution ellipsoid and the direction cosines of theanglesbetweenthegreatestofthemajorsemiaxesaandthecoordinateaxes

FIGURE33

2621ThepositionofapointK2relativetoapointKxisdeterminedonthebasisofmeasureddistancesD1andD2 fromapointA andof theangle in thehorizontalplane (seeFigure33)Findthecovariancematrixof theerrors in thedeterminationof thepositionof thepointK2relativetothepointifweknowthatthemeanerrorsmadeinthedeterminationofthedistanceareequaltoEDandthosemadeinthedeterminationoftheangleareequaltoEαThe measuring errors are mutually independent and they obey normaldistributionlawsAssumethatthealtitudeHof thepointAoverthehorizontalplaneK1BK2isknownexactly

2622SolveProblem2621withthehypothesisthatweknow(exactly)notthealtitudeHbuttheangle

1 If one chooses as conjugate directions the principal diameters of the ellipse (ellipsoid) thedegenerate distribution laws characterize the distributions of independent rectangular coordinates of arandompoint

2SincethaangleεissmallthedeviationalongthearcεDcanbereplacedbyadeviationofmagnitudeεDalongthetangentandonecanconsiderthisdeviationperpendiculartotheradiusD


27 THEENTROPYOFRANDOMEVENTSANDVARIABLES

BasicFormulasLetA1A2hellipAnbeacompletesetofmutuallyexclusiveeventsThentheentropyofthissetofeventsisdefinedas1

andrepresents theaveragequantityof informationreceivedbyknowingwhichof the eventsA1A2hellipAn occurred in a certain trial Thus the entropy is ameasureofuncertaintyarisingafterperformingtrialsinvolvingacompletesetofmutuallyexclusiveeventsA1A2hellipAn

A similar formula defines the entropy H[X] of a discrete variable Xassumingvaluesx1x2hellipxnwithprobabilitiesp1p2hellippn

Thesameformulasholdforn=infinThemeasureofuncertaintyofa randomvariableX assuminga continuous

series of values and having a given probability density f(x) is the differentialentropyH[X]definedbytheformula

wheref(x)logaf(x)=0forthosevaluesofxforwhichf(x)=0The conditional entropy of a random variableX with respect to a random

variableYisdefinedby

fordiscreteXandYandforcontinuousXandYbytheconditionaldifferentialentropy

WecalltheexpectationoftheconditionalentropytheconditionalmeanentropyHy[X]Fordiscreterandomvariables

andforcontinuousrandomvariables

SimilarformulasholdforsystemsofrandomvariablesForexample

representstheentropyofasystemofnrandomvariables

theconditionalmeanentropyofthesubsystem(XY)withrespecttoZand

theconditionalmeanentropyof therandomvariableZwith respect to randomvariablesXandYWealsohavetheinequalities

and

inwhichequalitycorrespondstothecaseofindependentrandomvariablesFor a = 2 the unit of measure for entropy represents the entropy of a

complete set of twomutually exclusive equally possible eventsForane2 thevalueoftheentropycomputedfora=2mustbemultipliedbyloga2Theunitofmeasureforentropyiscalledbinaryfora=2decimalfora=10andsoon


Example271Anumberof shots are firedat two targets two shots at thefirsttargetandthreeatthesecondoneTheprobabilitiesofhutingatargetinoneshot are equal to 12 and 13 respectivelyWhich of the two targets yields amorecertainoutcome

SOLUTIONTheoutcomeisdeterminedby thenumberofhitsscoredwhich

obeysthebinomialdistributionlawWeformthedistributionseriesofthefirsttargetforn=2andp=12(Table

16)andofthesecondtargetforn=3p=13(Table17)TABLE16

TABLE17

The entropy of the number of hits is a measure of the uncertainty of theoutcomeForthefirsttargetwehave

andforthesecondone

TheoutcomeinthecaseofthefirsttargethasagreatercertaintySimilarlyonecansolveProblems271to2711

Example272AmongalldistributionlawsofacontinuousrandomvariableX with the same known variation D find the distribution with the maximaldifferentialentropy

SOLUTION According to a theorem in calculus of variations to find afunctiony=y(x)thatrealizesanextremumoftheintegral

underconstraints

itisnecessarytosolvetheEulerequation

where and constants λs are found from the givenconstraintsInourexamplewearelookingforthemaximumoftheintegral

undertheconstraints

and


Consequentlytheequationforf(x)hastheform

andtherfore

where

Fromtheconstraintswefindthat

ThesolutionobtainedcorrespondstomaximalentropyTherefore for a given variation D the maximal entropy has the normal

distributionlaw

Problems2712to2715canbesolvedinasimilarmanner

Example273Provethatthemaximalentropyofadiscreterandomvariableis logan(nbeing thenumberofpossiblevaluesof therandomvariable)and isattainedforp1=p2=hellip=pn=1n

SOLUTIONWeshallmakeuseoftheinequalityInxge1ndash1x(xgt0)(equalityoccursonlyforx=1)Applyingthisinequalityweobtain

Itfollowsthat

Tothecasenpk=1therecorrespondsmaximalentropyloganOnecansolveProblem2716similarly

PROBLEMS

271Twournscontain15ballseachThe firsturncontains five red sevenwhite and three black balls the second urn contains four red four white andsevenblackballsOneballisdrawnfromeachurnFindtheurnforwhichtheoutcomeoftheexperimentismorecertain

272Theprobabilityofoccurrenceofaneventispandofnonoccurrenceq= 1 ndash p For which value of p does the result of the trial have the maximaluncertainty

273Forwhichofthefollowingtwoexperimentsdoestheoutcomehavethegreatestuncertainty(a)arandompointistakeninsideanequilateraltriangleandldquosuccessrdquomeansthepointlandsinsidetheinscribedcircle(b)arandompointistaken inside a circle and ldquosuccessrdquo means the point lands inside a givenequilateraltriangleinscribedinthecircle

274Byjoiningthemidpointsofadjacentsidesofaregularn-polygononeconstructsanotherregularn-polygoninscribedinthefirstApoint takeninsidethefirstpolygonmayturnouttobeinsideoroutsidetheinscribedpolygon

Find (a) the entropy of the experiment (b) the value of n for which theentropyismaximal

275TheprobabilityforrealizationofaneventAatonetrialispThetrialsarerepeateduntilAoccursforthefirsttimeFindtheentropyofthenumberoftrialsandclarifythecharacterofvariationoftheentropywiththechangeofp

276 Determine the entropy of a random variable obeying a binomialdistributionlaw(a)inthegeneralcase(b)forn=2p=q=05

277Determine theentropyofacontinuous randomvariableobeying (a) auniformprobabilitydistributionovertheinterval(cd)(b)anormaldistributionlawwithvariance (c)anexponentialdistributionoftheform

278FindtheentropyofarandomvariableXwithadistributionfunction

279EstimatetheconditionaldifferentialentropyH[X|y]andtheconditionalmeandifferentialentropyHy[X]ofarandomvariableXwithrespecttoYandalsoH[Y | x] andHX[Y] of the random variable Y with respect to X for thesystemofnormalrandomvariables(XY)

2710Findtheentropyofasystemofnrandomvariablesobeyinganormaldistributionlaw

2711GiventheentropiesH[X]andH[Y]oftworandomvariablesXandYandtheconditionalmeanentropyHy[X]oftherandomvariableXwithrespecttoYfindtheconditionalmeanentropyHX[Y]ofYwithrespecttoX

2712AmongalldistributionlawsofacontinuousrandomvariableXwhoseprobability density vanishes outside the interval a lt x lt b determine thedistributionlawwithmaximaldifferentialentropy

2713AmongalldistributionlawsofacontinuousrandomvariableXwhoseprobabilitydensityvanishes forx lt 0 for a known expectationM[X] find thedistributionlawwithmaximaldifferentialentropy

2714 Find the probability density for which the differential entropy of arandomvariableismaximalifitssecondmomentism2

2715 Among all the distribution laws for continuous systems of randomvariableswithaknowncovariancematrixfindthedistributionlawforwhichtheentropyofthesystemismaximal

2716AmessageisencodedbyusingtwogroupsofsymbolsThefirstgrouphasksymbolswithprobabilitiesofoccurrencepllpl2hellipp1kthesecondgrouphasnsymbolswithprobabilitiesofoccurrencep21p22hellipp2n

For a fixedvalueof a find theprobabilitiesp1i andp2jcorrespondingtothemaximalentropy

2717 Experiment A consists of selecting an integer from 1 to 1050 atrandom experiment B of communicating the values of the remainders upondividingtheselectednumberby5and7FindtheentropyofexperimentAandtheconditionalmeanentropyofAwithrespecttoexperimentB

2718Betweentwosystemsofrandomvariables(X1X2hellipXn)and(Y1Y2hellipYn) there exists aone-to-onecorrespondenceYk =φk(X1X2hellipXn)Xk =Ψk(Y1Y2hellipYn)wherek=12hellipnFindtheentropyH[Y1Y2hellipYn]iftheprobabilitydensityfx(x1x2hellipxn)isknown

2719Twosystemsofrandomvariables(X1X2hellipXn)and(Y1Y2hellipYn)arerelatedbylinearexpressions

Evaluatethedifferenceoftheentropies

(a)inthegeneralcase(b)forn=3andthetransformationmatrix

28 THEQUANTITYOFINFORMATION

BasicFormulasThequantityofinformationobtainedintheobservationofacompletesetof

mutually exclusive events is measured by its entropy H the quantity ofinformation that can be obtained by observing the value of a discrete randomvariableXismeasuredbyitsentropyH[X]

ThequantityofinformationaboutarandomvariableXthatcanbeobtainedbyobservinganotherrandomvariableY ismeasuredbythedifferencebetweentheentropyofXanditsconditionalmeanentropywithrespecttoY

Fordiscreterandomvariables

IfafterreceivingamessageaboutthediscreterandomvariableY thevalueoftherandomvariableXiscompletelydefinedthenHy[X]=0andIy[X]=H[X]

IfXandYareindependentthenHy[X]=H[X]andIy[X]=0Forcontinuousrandomvariables

FromthesymmetryoftheformulasdefiningthequantityofinformationwithrespecttoXandYitfollowsthat


Example281UsingthemethodofShannon-Fano2encodeanalphabetthatconsistsoffoursymbolsABCandDiftheprobabilitiesofoccurrenceofeachsymbolinamessageare

Findtheefficiencyofthecodeiethequantityofinformationpersymbol

SOLUTIONWeorderthesymbolsofthealphabetaccordingtothedecreasingprobabilitiesofCABDandthendividethemsuccessivelyintogroups

In the firstdivision the firstgroupcontainsCand thesecondAB andDsinceP(C)=048andP(A+B+D)=052Weassignthecodedsymbol1tothe

firstgroupandtothesecond0SimilarlyfromthesecondgroupweobtainthesubgroupsAandB+Dwithprobabilities028and024andwiththecodes01and00FinallythegroupB+DisdividedintoBandDwithprobabilities014and010andcodes001and000

ItisconvenienttorepresentthecodingprocessbyTable18

TABLE18

Acompletesetofmutuallyexclusiveeventscorrespondstotheoccurrenceofone symbol of the alphabet and the total quantity of information in thisparticular example is the entropy of the alphabet Therefore the quantity ofinformation per coded symbol (efficiency of the code) equals the ratio of theentropyofthealphabettotheexpectedlengthofthecodedversionsofsymbols


Example282Theprobabilities fora signal tobe receivedornot receivedareαand respectivelyAsaresultofnoiseasignalenteringthereceivercanberecordedat itsoutputwithprobabilityβandnot recordedwith

probability In the absence of the signal at the input it can berecorded at the output (because of noise) with probability γ and not recordedwithprobability Whatquantityofinformationaboutthepresenceofthesignalattheinputdoweobtainbyobservingitattheoutput

SOLUTION Let X denote the random number of input signals and Y the

randomnumberofoutputsignalsThen

Thisimpliesthat

Onecanalsousetheformula

wheretheunconditionalentropyis

andtheconditionalmeanentropyis

Example 283 There are 12 coins of equal value however one coin iscounterfeitdifferingfromtheothersbyitsweightHowmanyweighingsusingabalancebutnoweightsarenecessaryinordertoidentifythecounterfeitcoinandtodeterminewhetheritislighterorheavierthantherest

SOLUTIONAnyof the12 coinsmay turnout tobe the counterfeit one andthusmaybe lighterorheavier thanagenuinecoinConsequently thereare24possible outcomes that for equal probabilities of these outcomes give as theentropyforthewholeexperimentusedtoidentifythecounterfeitcointhevaluelog224=3+log23=3+04770301=458

Eachweighingprocesshas threeoutcomeswhichunder theassumptionofequalprobabilitiesgiveanentropyequaltolog23=158

Therefore the minimal number of weighings cannot be smaller thanlog224log23=458158=290ieitisatleastthreeInfactitwillbeshownthatforanoptimalplanningoftheexperimentexactlythreeweighingswillbenecessary

Inorderthatthenumberofweighingsistheminimumeachweighingmustfurnishthemaximalquantityofinformationandforthispurpose theoutcomeofaweighingmusthavemaximalentropy

SupposethatinthefirstweighingthereareicoinsoneachofthetwopansAsmentionedpreviouslyinthiscasethreeoutcomesarepossible

(1)thepansremaininequilibrium(2)therightpanoutweighstheleft(3)theleftpanoutweighstheright

For the first outcome the counterfeit coin is among the 12 ndash 2i coins putasideandconsequentlytheprobabilityofthisoutcomeis

ForthesecondandthirdoutcomesthecounterfeitcoinisononeofthepansThustheprobabilitiesoftheseoutcomesare

In order that a weighing give the maximal information the probabilitydistributionof theoutcomesmusthavemaximalentropywhichmeans thatallprobabilitiesmustbeequalFromthisitfollowsthat

ieinthefirstweighingprocessfourcoinsshouldbeplacedoneachpanNext we consider separately the following two cases (a) in the first

weighing the pans remain in equilibrium (b) one of the pans outweighs theother

Incase(a)wehaveeightgenuinecoinsandfoursuspectcoinsthatarenotusedinthefirstweighingForthesecondweighingwecanplaceisuspectcoinsontherightpan(ile4)andjleisuspectandindashjgenuinecoinsontheleftpanInthiscasei+jle4sincethenumberofsuspectcoinsis4Allpossiblevaluesfori and j and the corresponding probabilities of the outcomes in the secondweighingincase(a)areincludedinTable19

TABLE19

InthistabletheentropyoftheexperimentisalsogivenItis

Themaximalentropyisgivenbyexperiments4and7Thus thereare twoequivalentversionsof the secondweighing it isnecessaryeither toplace twosuspect coinsononepan and on theother one suspect andonegenuine coin(experiment 4) or to place three suspect coins on one pan and three genuinecoinsontheother(experiment7)

Inbothversions thethirdweighingsolvestheproblemthat is it identifieithecounterfeitcoinanddetermineswhetheritislighterorheavierthantherest

Incase(b)inwhichoneofthepansoutweighstheotherinthefirstweighingthecoinsaredividedintothefollowingthreegroupsfoursuspectcoinswhiclareplacedon the rightpan four suspectcoinson the leftpan (4ldquorightrdquoam4ldquoleftrdquo)andfourgenuinecoinswhicharenotusedinthefirstweighing

If in the secondweighing one places i1 ldquorightrdquo and i2 ldquoleftrdquo coins on therightpanldquorightrdquoj2ldquoleftrdquoandi1+i2ndashj1ndashj2genuinecoinsonthelefpanandthencomparestheentropyofallthepossibleversionstherewillbe1equivalentversionswithmaximal(equal)entropyAnyoftheseversionsfoexamplei1=3i2=2j1=1j2=0ori1=1i2=2j1=0j2=2givesmaximalinformationandpermitsustoidentifythecounterfeitcoininthithirdweighingandtofindoutwhetheritislighterorheavierthantherest


PROBLEMS

281 A rectangle is divided into 32 squares by four vertical and eighthorizontal lines A point can be inside any one of these squares with equalprobability

Find the quantity of information in the messages that (a) the point is insquare27(b)thepointliesinthethirdverticalandthefirsthorizontalline(c)thepointliesinthesixthhorizontalline

282ThereareN coinsof equalvalueofwhichone is counterfeit that islighterthantherest

How many weighings on a balance without weights are necessary toidentifythecounterfeitcoinWhatisthemaximalNforwhichfiveweighingsaresufficient

283 The symbols of the Morse Code can appear in a message withprobabilities051foradot031foradash012foraspacebetweenlettersand006 fora spacebetweenwordsFind theaveragequantityof information ina

messageof500symbolsifthereisnorelationbetweensuccessivesymbols284AcompositesystemcanbeinoneofNequallyprobablestatesAjThe

stateofthesystemcanbedeterminedbyperformingsomecontrolexperimentstheresultofeachshowingthegroupofstatesinwhichthesystemcanbe

InoneoftheexperimentsasignalisobservedinthestatesAlA2hellipAkandnotobservedinstatesAk+1Ak+2hellipANInanotherexperiment thesignal isobservedifthesystemisinoneofthestatesA1A2hellipAl(llek)orAk+1Ak+2hellipAk + r (r leN ndash k) and not observed in the rest What is the quantity ofinformationinthefirstandsecondexperiments

285Adefective televisionsetcanbe inoneoutof fivedifferent states towhichtherearecorrespondingdifferenttypesoffailuresToidentifythetypeoffailureoneperformsseveraltestsoutofatotalofsevenpossibletestswhichfordifferentstatesofthetelevisionsetmakeacontrollightbulbturnonoroffInthefollowingtablethesestatesaredenotedbyonesandzeros

Find a sequence consisting of the minimal number of tests that permitdeterminationofthetypeoffailure

286 Somemessages use the symbols of the alphabetA1A2A3 A4 withprobabilitiesP(A1)=045P(A2)=010P(A3)=015P(A4)=030

To transmit amessage througha communicationchannel onecanuse twocodes 1 and2 In the first code the symbolsabc andd and in the secondcodethesymbolsadbandccorrespondtothesymbolsofthealphabet

Determine the efficiency of the codes ie the average quantity ofinformationtransmittedpertimeunitifthetransmissiontimesofthesymbolsof

thecodethroughthecommunicationchannelforconventionaltimeunitsare

287 Under the assumptions made in the preceding problem along withcodes1and2considerotherpossiblecodesandfindthemostefficientone

288 For the transmission of some messages one uses a code of threesymbols whose probabilities of occurrence are 08 01 and 01 There is nocorrelation among the symbols of the code Determine the redundancy of thecodethatisthedifferencebetween1andtheratiooftheentropyofthegivencodetothemaximalentropyofacodecontainingthesamenumberofsymbols

289 A message consists of a sequence of two letters A and B whoseprobabilitiesofoccurrencedonotdependontheprecedingletterandareP(A)=08P(B)=02

Perform the codingbyusing themethodofShannon-Fano for (a) separateletters(b)blocksconsistingoftwo-lettercombinations(c)blocksofthree-lettercombinations

Comparethecodesaccordingtotheirefficiency2810 Compare the codes of the preceding problem according to their

redundancybycalculatingthemeanprobabilitiesofoccurrenceofthesymbolajbytheformula

whereZijisthenumberofsymbolsajintheithcodedcombinationandRiisthenumberofallsymbolsintheithcombination

2811 A message consists of a sequence of letters A B and C whoseprobabilities of occurrence do not depend on the preceding combination oflettersandareP(A)=07P(B)=02andP(C)=01

(a)Perform thecodingby themethodofShannon-Fanoforseparate lettersand two-letter combinations (b) compare the efficiencies of the codes (c)comparetheredundanciesofthecodes

2812 The probabilities of occurrence of separate letters of the Russianalphabet are given in Table 20 where the symbol ldquomdashrdquo denotes the spacebetweenwords

Perform thecodingof thealphabetby themethodofShannon-Fano if theprobability of occurrence of a letter is independent of the occurrences of the

precedingletters

TABLE20

2813 An alphabet consists of n symbols Aj (j = 1 2 hellip n) whoseoccurrencesinamessageareindependentandhaveprobability

wherekjarepositiveintegersand

Showthat ifonecodesthisalphabetbythemethodofShannon-Fanoeachcoded symbol contains amaximalquantityof information equal toonebinaryunit(onebit)

2814 Two signals A1 and A2 are transmitted through a communicationchannelwiththeprobabilitiesP(A1)=P(A2)=05Attheoutputofthechannelthesignalsare transformed intosymbolsa1anda2 and as a result of noise towhichA1andA2aresubjectedequallyerrorsappear intransmissionsothatanaverageofonesignaloutof100isdistorted(a1becomesa2ora2becomesa1)

Estimate the average quantity of information per symbolCompare itwiththequantityofinformationintheabsenceofnoise

2815SignalsA1A2hellipAnaretransmittedwithequalprobabilitiesthroughacommunicationchannelIntheabsenceofnoisethesymbolajcorrespondstothesignalAj(j=12hellipm)Inthepresenceofnoiseeachsymboliscorrectlyreceivedwithprobabilitypandisdistortedtoanothersymbolwithprobabilityq=1ndashpEvaluatetheaveragequantityofinformationpersymbolinthecasesofabsenceandofpresenceofnoise

2816 Signals A1 A2 hellip Am are transmitted through a communicationchannelwithequalprobabilitiesIntheabsenceofnoisethesymbolcorrespondstothesignalAj(j=12hellipm)BecauseofthepresenceofnoisesignalAjcanbereceivedcorrectlywithprobabilityPjjorassymbolaiwithprobabilitypij(ij=1

2hellipm pij=1)Estimatetheaveragequantityofinformationpersymbolthat is transmitted through the channel whose noise is characterized by thematrix||pij||

1p(Aj)istheprobabilityofeventAjp(Aj)logap(Aj)=0ifp(Aj)=02 In the case of encoding by the method of Shannon-Fano a collection of symbols(alphabet)

originallyorderedaccording to thedecreasingprobabilitiesofoccurrenceof the symbols isdivided intotwogroupssothatthesumsoftheprobabilitiesofthesymbolsappearingineachgroupareapproximatelyequalEachofthegroupsisthensubdividedintotwosubgroupsbyusingthesameprinciple theprocesscontinuesuntilonlyonesymbolremainsineachgroupEachsymbolisdenotedbyabinarynumberwhosedigits(zerosandones)showtowhichgroupagivensymbolbelongsinaparticulardivison

VI THELIMITTHEOREMS

29 THELAWOFLARGENUMBERS

BasicFormulas

IfarandomvariableXhasafinitevariancethenforanyεgt0Chebyshevrsquos

inequalityholdsIfX1X2hellipXnhellipisasequenceofrandomvariablespairwiseindependent

whosevariancesareboundedbythesameconstantD[Xk]leCk=12hellipthenfor any constant ε gt 0

(Chebyshevrsquostheorem)If therandomvariablesX1X2hellipXnhellipallhave thesamedistributionand

have finite expectations then for any constant ε gt 0

(Khinchinrsquostheorem)ForasequenceofdependentrandomvariablesX1X2hellipXnhellipsatisfying

theconditionforanyconstantεgt0wehave

(Markovrsquostheorem)

In order that the law of large numbers be applicable to any sequence ofdependentrandomvariablesX1X2hellipXnhellipieforanyconstantεgt0forthe

relationtobefulfilleditisnecessaryandsufficientthatthefollowingequalityholdtrue


Example291Provethatifφ(x)isamonotonicincreasingpositivefunction

andM[φ(X)]=mexiststhenSOLUTIONTaking intoaccount thepropertiesofφ(x)weobtain a chainof

inequalities

since ThisimpliesthatP(Xgtt)lemφ(t)whichwewishtoproveSimilarlyonecansolveProblems292to295

Example292GivenasequenceofindependentrandomvariablesX1X2hellip

XnhellipwiththesamedistributionfunctiondeterminedwhetherKhinchinrsquostheoremcanbeappliedtothissequence

SOLUTION For the applicability ofKhinchinrsquos theorem it is necessary that

the expectation of the random variableX exist ie

converge absolutely However

ietheintegraldoesnotconvergetheexpectationdoesnotexistandKhinchinrsquostheoremisnotapplicable

Example 293 Can the integral afterthe change of variables y = ax be calculated by a Monte-Carlo method

accordingtotheformulawhereykarerandomnumbersontheinterval[01]

SOLUTION Performing the previously mentioned change of variables weobtain

ThequantityJncanbeconsideredanapproximatevalueofJonlyifthelimitequalitylimnrarrinfinP(|JnndashJ|ltε)=1holdstrue

The random numbers yk have equal distributions and thus the functions(1yk)sin(ayk)alsohaveequaldistributionsToapplyKhinchinrsquostheoremoneshouldmake sure that the expectationM[(1Y) sin (aY)] exists whereY is arandomvariableuniformlydistributedover the interval [0 1] ie one should

provethat (1y)sin(ay)convergesabsolutelyHoweverifwedenotebystheminimalintegersatisfyingtheinequalitysge

aπ then

Since

theintegraldivergestoo

ThelattermeansthatM[(1Y)sin(aY)]doesnotexistandconsequentlytheMonte-Carlomethodisnotapplicableinthisparticularcase

Example294Canthequantity

betakenasanapproximatevalueofthevariationoferrorsgivenbyadeviceifX1X2hellipXnhellipareindependentmeasurementsofaconstantquantityaandiftheyallhavethesamedistributionfunctions

SOLUTIONLetusdenotethetruevalueofthevariancebyσ2Thequantity can be considered as an approximate value for σ2 if

Since X1 X2 hellip Xn hellip are independent random variables with equaldistributions the variables Yk = (Xk ndash a)2 are independent and have equaldistributions

Wehave

where =M[Xk]TosatisfytheequalityM[Yk]=σ2itisnecessarythat =awhichmeansabsenceofsystematicerrorsinmeasurements

Thusifthemeasuringdevicedoesnotgivesystematicerrorstheconditionsfor applicability of the law of large numbers are satisfied and consequently

PROBLEMS

291Use Chebyshevrsquos inequality to estimate the probability that a normalrandomvariablewill deviate from its expectation bymore than (a) fourmeandeviations(b)threemean-squaredeviations

292 Prove that for any random variable X and any ε gt 0 the following

inequalityholdswhereJ=M[eεX]

293 Prove that if M[eaX] exists

294 A random variable X obeys the exponential distribution law

Provethatthefollowinginequalityholdstrue

295TheprobabilityofoccurrenceofaneventAinoneexperimentisfrac12Canoneassertthatwithprobabilitygreaterthan097thenumberofoccurrencesofAin1000independenttrialswillbewithinthelimitsof400to600

296IsthelawoflargenumbersvalidforthearithmeticmeanofnpairwiseindependentrandomvariablesXkspecifiedbythedistributionseriesinTable21

TABLE21

297LetXkbearandomvariablethatcanassumewithequalprobabilityoneof two values ks orndashks Forwhich value of s does the law of large numbersapply to thearithmeticmeanof the sequenceof independent randomvariablesX1X2hellipXkhellip

298Provethatthelawoflargenumbersisapplicabletothearithmeticmeanofasequenceof independentrandomvariablesXkspecifiedbythedistributionseriesincludedinTable22

TABLE22

299ArethesufficientconditionssatisfiedfortheapplicabilityofthelawoflargenumberstoasequenceofmutuallyindependentrandomvariablesXkwithdistributions specified by the formulas

2910TherandomvariablesX1X2hellipXnhelliphaveequalexpectationsandfinitevariationsIsthelawoflargenumbersapplicabletothissequenceifallthecovariances arenegative

2911 Prove that the law of large numbers is applicable to a sequence of

random variables inwhich each random variable can depend only on randomvariableswithadjacentnumbersandall the randomvariablescontained in thesequencehavefinitevariancesandexpectations

2912A sequenceof independent and equallydistributed randomvariablesX1 X2 hellip Xi hellip is specified by the distribution series

where is the value of the Riemannfunctionforargument3Isthelawoflargenumbersapplicabletothissequence

2913GivenasequenceofrandomvariablesX1X2hellipXnhellipforwhichDlecandrijrarr0for|indashj|rarrinfin(rij isthecorrelationcoefficientbetweenXiandXj) prove that the law of large numbers can be applied to this sequence(Bernsteinrsquostheorem)


determinewhetherthelawoflargenumbersappliestothissequence

30 THEDEMOIVRE-LAPLACEANDLYAPUNOVTHEOREMS

BasicFormulas

AccordingtothedeMoivre-Laplacetheoremforaseriesofn independenttrialsineachofwhichaneventAoccurswiththesameprobabilityp(0ltplt1)there obtains the relation

wheremisthenumberofoccurrencesofeventAinntrialsand

istheLaplacefunction(probabilityintegral)whosevaluesareincludedin8Tin

thetablelistonpage471According toLyapunovrsquos theorem fora sequenceofmutually independent

random variablesX1X2hellipXkhellip satisfying for some δ gt 0 the condition

thefollowingequalityholds

where is the expectation of is thevarianceofXk

To prove that Lyapunovrsquos theorem is applicable to equally distributedrandomvariablesitissufficienttoshowthatthevariancesofthetermsarefiniteanddifferentfromzero


Example301Theprobabilitythatanitemwillfailduringreliabilitytestsisp= 005What is the probability that during testswith 100 items the numberfailingwillbe(a)atleastfive(b)lessthanfive(c)betweenfiveandten

SolutionBythedeMoivre-Laplacetheorem

ifnissufficientlylargeByassumptionn=100p=005q=1ndashp=095

(a)Theprobabilitythatatleastfiveitemsfailis

(b)Theprobabilitythatlessthanfiveitemsfailis

(c)Theprobabilitythatfivetotenitemsfailis


Example302HowmanyindependenttrialsshouldbeperformedsothatatleastfiveoccurrencesofaneventAwillbeobservedwithprobability08iftheprobabilityofAinonetrialisP(A)=005

SOLUTIONFromthedeMoivre-Laplacetheoremweseethat

Forn=1wehaveΦ(436 )asymp1thereforesubstitutingP(mge5)=08we

obtain

or

From8T in the table list on page 471we find the argument x = ndash08416corresponding to the value of the functionΦ(x) = ndash06 Solving the equation

wefindtheuniquerootn=144ThusinorderthatAoccuratleastfivetimeswithprobability08144trialsarenecessary

FollowingthisexampleonecansolveProblems305to307

Example303Howmanytrialsshouldbeperformedtocalculatetheintegral

by a Monte-Carlo method so that with probability 09 the relative error incalculatingthevalueoftheintegralislessthan5

SOLUTIONThe integral canbe lookeduponas theexpectationofthefunctioncosxoftherandomvariableXuniformlydistributedover the interval (0 π2) Then the approximate value of the integral is

whereXkarerandomnumbersontheinterval(0π2)Letusformtherandomvariable

whichaccordingtoLyapunovrsquostheoremhasthedistributionfunction

because the variables cos Xk are independent and equally distributed with afinite variance different from zero and J = M[Jn] We have

ApplyingLyapunovrsquostheoremforb=ndasha=εweget

consequentlyitfollowsthatε=1645Inorderthattherelativeerror(JnndashJ)Jbelessthan005sinceJ=1 it is

necessarytoperformntrialssothatthusweobtainngt252


PROBLEMS

301Theprobabilityofoccurrenceofaneventinonetrialis03Whatistheprobability that therelativefrequencyof thisevent in100 trialswill liewithintherange02to04

302Thereare100machinesofequalpoweroperatingindependentlysothateachisturnedonduring08oftheentireoperatingtimeWhatistheprobabilitythatatanarbitraryinstantoftime70to86machineswillbeturnedon

303Theprobability thatacondenser failsduringa timeT is02Find theprobabilitythatamong100condensersduringtimeT(a)atleast20condensers(b)fewerthan28condensers(c)14to26condenserswillfail

304UsingthedeMoivre-Laplacetheoremshowthatforasufficientlylargenumber of trials

wheremn is the frequency of occurrence of the event whose probability ofoccurrenceisp

305TheprobabilityofaneventisevaluatedbyaMonte-CarlomethodFindthenumberofindependenttrialsthatinsurewithprobabilityatleast099thatthevalueoftherequiredprobabilitywillbedeterminedwithanerrornotexceeding

001ApplyChebyshevrsquosandLaplacersquostheorems306Theprobabilitythatanitemselectedatrandomisdefectiveineachtest

is01Alotisrejectedifitcontainsatleast10defectiveitemsHowmanyitemsshould be tested so that with probability 06 a lot containing 10 per centdefectiveitemswillberejected

307 How many trials are necessary so that with probability 09 thefrequencyofagiveneventwilldifferfromtheprobabilityofoccurrenceofthiseventbyatmost01iftheprobabilityoftheeventis04

308Theprobabilityofoccurrenceofacertaineventinonetrialis06Whatistheprobabilitythatthiseventwillappearinmostof60trials

309 The probability of eventA is 13 and 45 000 independent trials areperformedWhatisthemeandeviationEofthenumberofoccurrencesofeventAfromtheexpectationofthisnumber

3010Thecalculationoftheintegral ismadebyaMonte-Carlomethodbasedon1000independenttrialsEvaluatetheprobabilitythattheabsoluteerrorintheestimateofJwillnotexceed001

3011 How many trials should be performed to calculate the integral

byaMonte-CarlomethodsothatwithprobabilityPge099theabsoluteerrorofthecomputedvaluewillnotexceed01percentofJ

3012TheprobabilityP(C)=P(A+B)whereP(B )isknownisestimatedby aMonte-Carlomethod in twoways (1) the approximate value ofP(C) isfoundasthefrequencyofoccurrenceoftheeventCinaseriesofnindependenttrials (2) the frequency mn of occurrence of the event A in a series of nindependenttrialsisfoundandtheapproximatevalueofP(C)isevaluatedbythe

formula(a) Prove that both ways lead to the same result (b) find the necessary

numberof trials ineachcaseso that theerror in theestimateofP(C)doesnotexceed001withprobability095 ifP(B )=03andthevalueofP(A) isoforder04

3013Thereare100urnscontaining five redand95blackballs eachTheexperimentissuchthatafteraballisdrawnitisreturnedtothesameurnandtheoutcome of the trial is not communicated to the observer How many trialsshouldbeperformedsothat(a)theprobabilityis08thatatleastoneredballisdrawnfromeachurn(b)theprobabilityis08thatatleastoneredballisdrawn

fromatleast50urns3014 Compute the characteristic function EYn of the random variable

and find its limit for n rarr infin if the random variables X1 X2 hellip Xn hellip areindependent and have equal probability densities or distribution series of the

form3015Find the limit fornrarrinfinof the characteristic functionEYn(u) of the

randomvariableif the random variables X1 X2 hellip Xn hellip are independent have equaldistribution lawsexpectationsandvariancesand themomentsofhigherorderarebounded


31 GENERALPROPERTIESOFCORRELATIONFUNCTIONSANDDISTRIBUTIONLAWSOFRANDOMFUNCTIONS

BasicFormulasArandomfunctionofarealvariable t isafunctionX(t) that foreach t isa

random variable If the variable t can assume any values on some (finite orinfinite) interval thentherandomfunctioniscalledastochasticprocess if thevariabletcanassumeonlydiscretevaluesX(t)iscalledarandomsequence

The(nonrandom)function whichforeachtistheexpectationM[X(t)]of the random variableX(t) is called the expectation of the random functionX(t)

The correlation (autocorrelation) functionKx(t1 t2) of the random functionX(t) is defined by the formula

wheredenotesthecomplexconjugate1Forstationaryrandomfunctionswehave

ThevarianceoftheordinateofarandomfunctionisrelatedtoKx(t1t2)by the

formulaD [X(t)] = =Kx(t1 t2) The normalized correlation function is

definedbytheformulaThe total character of a random function is given by the collection of

distributionlaws

wheref(x1hellipxn|t1helliptn)isthedensityofthejointdistributionofthevalues

of the random function at times (t1 t2 t3hellip tn) The expectation andcorrelationfunctionKx(tlt2)areexpressedintermsofthefunctionsf(x1|t1)andf(x1x2|t1t2)bytheformulas(forcontinuousrandomfunctions)2

For a normal stochastic process the joint distribution at n times iscompletelydefinedbythefunctions andKx(t1t2)bytheformulasforthedistribution of a system of normal random variables with expectations

andwhoseelementsofthecovariancematrixarekjl=Kx(tjtl)lj=12hellipnThemutualcorrelationfunctionRxy(t1t2)oftworandomfunctionsX(t)and

Y(t) is specified by the formula

Forstationaryprocesses

The notion of correlation function extends to random functions of severalvariables If for example the random functionX (ξ η) is a function of twononrandom variables then


TheproblemsofthissectionareoftwotypesThoseofthefirsttypeaskforthecorrelation functionofa randomfunctionand for thegeneralpropertiesofthecorrelationfunctionInsolvingtheseproblemsoneshouldstartdirectlyfromthedefinitionof thecorrelation functionTheproblemsof thesecond typeaskfortheprobabilitythattheordinatesofarandomfunctionassumecertainvaluesTo solve these problems it is necessary to use the corresponding normaldistributionlawspecifiedbyitsexpectationandcorrelationfunction

Example 311 Find the correlation function Kx(t1 t2) if

whereωjareknownnumberstherealrandomvariablesAjandBjaremutuallyuncorrelatedandhavezeroexpectationsandvariancesdefinedbytheequalities

SOLUTIONSince bythe definition of the correlation function

IfweopentheparenthesesandapplytheexpectationtheoremwenoticethatallthetermscontainingfactorsoftheformM[AjAl]M [BjBl] for jne landM

[AjBl]foranyjandlarezeroand ThereforeKx(t1

t2)= cosω(t2ndasht1)SimilarlyonecansolveProblems313to316and3110

Example 312 LetX(t) be a normal stationary random function with zeroexpectationProvethatif

then

wherekx(τ)isthenormalizedcorrelationfunctionofX(t)

SOLUTIONUsingthefactthatX(t)isnormalweseethatthedistributionlawofsecondordercanberepresentedas

Therequiredexpectationcanberepresentedintheform

Since (12)[1 + (x1x2|x1x2|)] is identically equal to zero if the signs ofordinates x1 and x2 are different and equal to one otherwise we see that

which by integration leads to the result mentioned in the Example (Forintegrationitisconvenienttointroducenewvariablesrφsettingx1=rcosφx2=rsinφ)

PROBLEMS

311Provethat

(a)|Kx(t1t2)|leσx(t1)σx(t2)(b) 312Provethat|Rxy(t1t2)|leσx(t1)σy(t2)313Prove that thecorrelation functiondoesnotchange ifanynonrandom

functionisaddedtoarandomfunction314 Find the variance of a random function X(t) whose ordinates vary

stepwisebyquantitiesΔj at random timesThenumberof steps during a timeintervalτobeysaPoissondistributionwithaconstantλτandthemagnitudesofthe steps Δ are mutually independent with equal variances σ2 and zeroexpectationsandX(0)isanonrandomvariable

315Find thecorrelation functionof a random functionX(t)assuming twovalues +1 and ndash 1 the number of changes of sign of the function obeys a

Poisson distribution with a constant temporal density λ and can beassumedzero

316ArandomfunctionX(t)consistsofsegmentsofhorizontallinesofunitlengthwhoseordinatescanassumeeithersignwithequalprobabilityandtheirabsolute values obey the distribution law

EvaluateKx(τ)317ThecorrelationfunctionoftheheelangleofofashipΘ(t)hastheform

Findtheprobabilitythatattimet2=t1+τtheheelangleΘ(t2)willbegreaterthan15degifΘ(t)isanormalrandomfunction =0Θ(t1)=5degτ=2seca=30deg2α=002secndash1andβ=075secndash1

318 It ispossible tousea sonicdepth finderona rollingshipwhoseheelangleΘ(t)satisfies|Θ(t)|leθ0ThetimeforthefirstmeasurementisselectedsothatthisconditionissatisfiedFindtheprobabilitythatthesecondmeasurementcanbeperformedafterτ0secifΘ(t) isanormalfunction =0 thevariance

and the normalized correlation function k(τ)=Kθ(τ) areknown

319ThecorrelationfunctionoftheheelangleΘ(t)ofashipisKθ(τ)=aendashα|τ|

[cosβτ+(αβ)sinβ|τ|]wherea=36deg2α=025secndash1andβ=157secndash1Attimettheheelangleis2degΘ(t)ge0Findtheprobabilitythatattime(t+2)second the heel angle will have an absolute value less than 10deg if Θ(t) is anormalrandomfunctionand (t)=0

3110 Find the expectation and variance of the random function Y(t) =a(t)X(t) + b(t) where a(t) and b(t) are numerical (nonrandom) functions andKx(t1t2)and areknown

3111 Find the distribution law of first order for the values of the randomfunction

ifthedistributionlawsoffirstorderfortherandomfunctionsA(t)andΘ(t)havetheform

where ω is a constant and at the same timeA(t) and Θ(t) are mutuallyindependent

3112RandompointsaredistributedovertherealaxissothattheprobabilityPnofoccurrenceofnpointsonaprescribedintervalτisgivenbyPoissonrsquoslawPn=(λτ)nnendashλτwhereλisapositiveconstantFindthedistributionlawoffirstorderforarandomfunctionX(m)definingthedistancebetweenthemthandthe(m+n+1)strandompoints

3113Find thedistribution law for thevaluesofa randomfunctionof twovariablesU(xy)ifandthecorrelationfunctionKζ(ξη)definedby

isgivenintheform

whereζ(ξη)isanormalrandomfunctiona=100α1=02α2=01β1=05β2=10ξ0=1andη0=2

32 LINEAROPERATIONSWITHRANDOMFUNCTIONS

BasicFormulasAn operator is amapping of functions into functions3 The operator L0 is

called linear and homogeneous if it fulfills the conditions

whereAisanyconstantandφ(t)φ1(t)andφ2(t)areanyfunctionsA linear nonhomogeneous operator L is any operator related to a linear

homogeneousoperatorL0bytheexpression

whereF(t)issomefixedfunctionIfY(t)=L0X(t)andtheoperatorL0islinearandhomogeneousthen

whereListheoperatorLinwhichallcoefficientshavebeenreplacedbytheircomplexconjugatestheindicest1andt2inthenotationoftheoperatorL0showthat in the firstcase theoperatoractsonvariable t1 and in the secondon thevariable t2 (The possibility of applying the operator to the given randomfunction should be verified in each concrete case) If L is a nonhomogeneousoperatorcorrespondingtothehomogeneousoperatorL0andtothefunctionF(t)and if Z(t) = LX(t) then

iethecorrelationfunctiondoesnotdependonF(t) thefunctionengenderingthenonhomogeneityoftheoperatorL

A random function is differentiable (once) if its correlation function has asecondmixedpartialderivativeforequalvaluesof thevariableswhich in thecaseofstationaryfunctionsisequivalenttotheexistenceofasecondderivative

ofK(τ)forτ=0It is considerably more difficult to find the expectation and correlation

function for the result of the application of a nonlinear operator to a randomfunction whose probability properties are known An exceptional case isrepresented by a normal stochastic process for some types of nonlinearoperatorsFor example ifX(t) is a normal random function (we considerX(t)real) and Y(t) = X2(t) then

since theexpectationof theproductof fournormalvariablesX(t1)X(t1)X(t2)andX(t2)canbeobtainedbyadifferentiationofthecharacteristicfunctionofasystemofrandomvariables(seeSection23page124)

Inthesamewayonecanobtaintheexpectationandcorrelationfunctionofanessentiallynonlinearexpression

ifX(t)isnormal(seeExample322)


Theproblemsinthissectioncanbesolvedbyusingthegeneralformulaforthecorrelationfunctionoftheresultobtainedbyapplyingalinearoperatortoarandom function however in some problems it is more convenient to startdirectlyfromthedefinitionof thecorrelationfunctionThesecondwaycannotbe avoided if in addition to linear operators a given expression also containsnonlinear operatorsThe following are considered examples of applications ofbothmethods

Example321 Find the standard deviation of the angleΨ of rotation of adirection gyroscope after 10 minutes of rotation as a result of the randommomentM(t)appearingon theaxisof the innersuspension ring if the lawofvariationofΨ(t)canberepresentedby theequation (t)=M(t)Hwhere thekinetic moment H = 21105 g cm2sec2 and

SOLUTIONSincebyintegrationwehaveΨ(t)=1H M(t1)dt1 (the initialconditionsbythenatureoftheproblemarezero)ieΨ(t)andM(t)arelinearlyrelated for the correlation function KΨ(t1 t2)we obtain

andforthevariance

Since

thelastintegralcanbecalculatedbyintegrationbypartsleadingto

Example322FindthevarianceoftheangleΨ(t)ofrotationofadirectiongyroscope after T = 10 minutes of rotation if Ψ is defined by the equation

whereΘisanormalstationaryrandomfunctionwithacorrelationfunction

where =0andbHareconstants

SOLUTION Here besides the linear operations of integration anddifferentiation the given expression contains the nonlinear operation signum

Thus using the temporary notation we set Y(t) = sgn X(t)UsingthedefinitionofKy(τ)asthesecondcentralmixedmomentoftherandomvariables Y1 = sgn X(t) and Y2 = sgn X(t + τ) we obtain

wherethedistributionlawf(x1x2)isnormalSubstitutingthevalueofthisdistributionlawandchangingfromrectangular

coordinatesx1x2 topolarcoordinatesoneeasilycalculatesboth integralsand

obtains

wherethenormalizedcorrelationfunctionkx(τ)isgivenbytheformula

Therequiredvariance

Theproblemcanbesolvedbyanothermethod too Ifweuse the formula

andsetitintheinitialdifferentialequationthenafterweintegratewithrespecttotimeandestimatetheexpectationofΨ2(t)weobtain

whereE(u1u2) is thecharacteristicfunctionfor thesystemofnormalvariablesX(t1)andX(t2)

IfwesubstituteinthelastintegraltheexpressionforE(u1u2)andintegrateitthreetimeswefindforD[Ψ(t)]thesameexpressionasjustobtained

Example323Find theexpectationandcorrelation functionof the randomfunction

where a(t) and b(t) are given (numerical) functions X(t) is a differentiablerandomfunctionand Kx(t1t2)areknown

SOLUTIONThefunctionY(t)istheresultofapplicationofthelinearoperator[a(t)+b(t)ddt]totherandomfunctionX(t)ThereforetherequiredresultcanbeobtainedbyapplyingthegeneralformulasHoweverthesolutioncanbefoundmore easily by direct computation of and Ky(t1 t2) We have

PROBLEMS

321FindthecorrelationfunctionofthederivativeofarandomfunctionX(t)if

322Findthecorrelationfunctionandvarianceoftherandomfunction

ifKx(τ)=aendashα|τ|[cosβτ+(αβ)sinβ|τ|]323 Let X(t) be a stationary random function with a known correlation

functionFindthemutualcorrelationfunctionofX(t)anddX(t)dt324HowmanyderivativesdoesarandomfunctionX(t)withacorrelation

functionKx(τ)=σ2endashα2τ2have

325HowmanytimescanonedifferentiatearandomfunctionX(t)ifKx(τ)=σ2endashατ[1+α|τ|+(13)α2τ2]

326Uptowhatorderdo thederivativesofarandomfunctionX(t)exist ifthe correlation function has the form

327ArandomfunctionX(t)hasacorrelationfunction

Findthemutualcorrelationfunctionof

328ThecorrelationfunctionofarandomfunctionX(t)hastheform

findthevariancesforthefunctions

329Given thecorrelationfunctionKx(τ)of thestationaryrandomfunction

X(t)

findthecorrelationfunctionof

3210 Find the probability P that the derivative V of a normal stationaryfunction X(t) will have a value greater than b = msec if

wherea=4sqmα=1secndash1β=2secndash13211Given the expectations correlation functions andmutual correlation

functionbetweentworandomfunctionsX(t)andY(t)findtheexpectationsandthecorrelationfunctionoftherandomfunction

3212 Express in terms of the distribution laws of a system of n randomfunctionsXj(t) (j=12hellipn) theexpectationand thecorrelation functionof

3213ThecorrelationfunctionKx(τ)ofastationaryrandomfunctionX(t) isknown Find the correlation function of Y(t) if

3214ArandomfunctionX(t)hasthecorrelationfunction


3215GiventhecorrelationfunctionKx(τ)ofarandomfunctionX(t)findthe

varianceof3216AstationaryrandomfunctionY(t)isrelatedtoanotherfunctionX(t)by

FindthecorrelationfunctionofX(t)ifX(t)=0fort=0andKy(τ)isknown

3217FindthecorrelationfunctionofX(t)andY(t)= X(ξ)dξifKx(t1t2)isknown

3218FindthevarianceofY(t)fort=20secif

3219Findthecorrelationfunctionandtheexpectationof

if andKx(t1t2)areknownandtheconstantsa0a1andb1arereal3220 Find the mutual correlation function of Ryz(t1 t2) if

whereabcanddarerealconstants

3221 The speed of an airplane is estimated with the aid of a gyroscopic

integratorthatgivesanerrorHere θ(t) is the error in the stabilization of the axis of the integrator the

correlationfunctionis

andgistheaccelerationofgravityFindthemean-squareerrorintheestimateofthevelocityafter10hoursofflight(τisgiveninseconds)

3222ArandomfunctionΘisrealnormalandstationaryand =0Findthe

correlationfunctionwhereabandcarerealconstants

3223Theperturbationmomentactingontherotorofagyroscopeinstalledon a ship is expressed in terms of the heel angle Θ(t) and the angle of trimdifference Ψ(t) by the relation

FindthecorrelationfunctionM(t)ifKθ(τ)andKΨ(τ)areknownRθΨ(τ)equiv0andΘ(t)andΨ(t)arenormal

3224 Given that Kx(τ) = endash α2τ2 find the correlation function Ky(τ) if

3225Given

findthemutualcorrelationfunctionbetweenX(t)andd2X(t)dt23226GiventhecorrelationfunctionKx(τ)findKx(t1t2)ifY(t)=a(t)X(t)+

b(t)d2X(t)dt2wherea(t)andb(t)arenumerical(nonrandom)functions3227Let

IsthereafunctionX(ξ)differentfromzeroforwhichY(t)isastationaryrandomfunction

3228IsthefunctionZ(t)=X(t)+YstationaryinthebroadsenseifX(t)isastationaryrandomfunctionandYis(a)arandomvariableuncorrelatedwithX(t)(b)Y=X(t0)

3229 Find the variance of the error Y(t) of a nonperturbed gyro-inertialsystem after one hour of its operation if Y(t) is defined by the equation

where v = 124middot10ndash3 secndash1 is the frequency of Shuler and X(t) is theaccelerometer error which can be considered a stationary normal function of

time3230Theangulardeviationsαandβofafreegyroscopeusedasavertical

indicatoronarollingshiparedefinedapproximatelybythesystemofequations

where themomentsof inertia I1 I2 thekineticmomentof therotorHand thecoefficientsofdryfrictionk1andk2areconstantsand theheelangleΘ(t)andthe angle of trim differenceΨ(t) can be assumed to be two stationary normalfunctionsoftimewithknowncorrelationfunctions

FindD[α(t)]andD[β(t)]iftislarge

Hint Introduceanewfunction

q=HI2p=HI1andreplacesgn[ (t)]andsgn byintegralsasshowninExample322

3231FindthevarianceofthefunctionZ(t)definedbytheequation (t)+a2[1+Y(t)]Z(t)=X(t)Z(0)=0whereX(t)andY(t)areindependentstationarynormal functions with zero expectations and whose correlation functions are

known

33 PROBLEMSONPASSAGES

BasicFormulasApassage (time) at a given levela for a random functionX(t) is a time t

whensomegraphofthisfunctioncrossesthehorizontallineX=a(frombelow)Theprobabilitythatapassage(time)liesinaninfinitelysmalltimeinterval

dt around point t is p(a | t) dt the temporal probability density p(a | t) isexpressedintermsofthedifferentialdistributionlawf(xv|t)oftheordinateof

randomfunctionX(t)anditsderivative computedattimetby

The temporal probability density for the intercept of the random function(goingdown)atthelevelais

Fornormalfunctions

Fornormalstationaryfunctions

Theaveragenumberofpassages ofastationaryrandomfunctionperunittimeisp(a)

The average number of passages of a stationary function during a time

intervalTis =Tp(a)Theaverageduration ofapassageofastationaryfunctionis

wheref(x)istheprobabilitydensityfortheordinatesofthisrandomfunction

Forastationarynormalprocess

Similarformulasholdfornonstationaryprocesses

Theproblemoffindingtheaveragenumberofmaximaofarandomfunction(the passage of the first derivative through zero from above) and some otherproblemscanbereducedtoproblemsonpassagesForasmallaveragenumberofpassagesduringatimeintervalTtheprobabilityQfornonoccurrenceofanyrun during this interval can be estimated approximately by the formula

ie the number of passages in the given interval can beconsideredasobeyingapproximatelyaPoissonlaw

The formulas for the average number of passages and the average timebetweensuccessivepassagescanbegeneralizedforrandomfunctionsofseveralvariables


Example331EvaluatetheaveragenumberoftimesduringT=10minutesin which the heel angle Θ(t) of a ship vanishes if = 0

whereτisexpressedinsecondsandΘ(t)isanormalrandomfunction

SOLUTIONTheaveragenumberofpassagesthroughzerois

Since

wehave

and thenumberof passagesduring10minutes =600middot01124=675The

requirednumberis2 =135

Example332TheheelangleΘ(t)andtheangleoftrimdifferenceΨ(t)areuncorrelatednormalrandomfunctionswhosecorrelationfunctionsaregivenbythe formulas

whereτisexpressedinsecondsandtheexpectations and areequaltozeroFindtheaveragetimethatthemastoftheshipisoutsidetheconewhoseaxis

isverticalandwhosegeneratingangleis2degifthedeviationofthemastfromthe

verticalvcanbedefinedbytheapproximateformula

SOLUTIONThiscasediffersfromtheprecedingonebecausethefunctionv(t)is not normal Therefore one should apply the general formula

wherev(t)=dv(t)dtTofindtheprobabilitydensityf(v)itisnecessarytointegratetheprobability

density of the systemof normal randomvariablesΘ(t)Ψ(t) over the domain

which can be performed easily if we pass

fromrectangularcoordinatesθψtopolarcoordinates φ=arctan(ψθ)

Afterintegrationweobtain

where I0(z) are theBessel functions of first kind of an imaginary variableToobtainf(vυ)itisnecessarytointegratetheprobabilitydensityofthesystemof

mutually independent random variables over thedomain of variance of its arguments where the following conditions hold

This integrationcanbeperformedeasily ifonepasses fromθ to thevariables Using the Jacobian of the transformation weobtain

By assumptions deg2sec2 and consequently thedouble integral is simplified and can be computed

Then

If we substitute the result obtained and the probability density f(v) in the

formulafor wegetSinceinthetheoryofBesselfunctionsitisprovedthat

theintegralinthenumeratorcanberepresentedas

InthelastintegralthevalueoftheargumentoftheBesselfunctionfortheupperlimit is very smallTherefore using the expansionof theBessel function in aseries

weobtain

thatis

Example 333 Find the average number of maxima of a normal randonfunctionX(t)perunittimeif

SOLUTIONTherandomfunctionX(t)hasamaximumif itsderivativehas a passage through zero from above that is

PROBLEMS

331Findtheaveragedurationof thepassageofanormalrandomfunctionX(t)throughthelevela=2cmif =ndash8cmandKx(τ)=100endash01|τ|(1+01|τ|)sqcmwhereτisexpressedinseconds

332 The average number of passages of a normal stationary functionthroughthelevela= inonesecond is001Find thevarianceof therateofchangeofthisfunctionifthevarianceofthefunctionitselfis64cm2

333Thecorrelation functionof theheelangleΘofa ship isgivenby theformula

Iftheprocessofrollingisnormalestimatetheaveragenumberoftimesin20minutesduringwhichtheheelangleisoutsidetheboundsplusmn25degif =0b=100deg2α=01secndash1andβ=07secndash1

334 The output errors of a dynamical system are normal with zero

expectationandcorrelationfunctionwherea = 5 square angularminutes andα = 15 secndash1 Estimate the averagenumber of times in which the system will be turned off if this occursautomaticallyinthecaseofanerrorwhoseabsolutevalueexceeds3prime

335Thecorrelationfunctionofanormalstochasticprocessis

Evaluate the time t at which the average number of passages through thelevela= perunittimeislessthanaprescribednumberp0(p0gtα2π)

336 To remove the damage caused by a random exterior perturbationcharacterized by a normal random functionX(t) it is necessary to use power

W(t)proportionalto Estimatetheaveragenumberoftimesperunittimeinwhichthepowerofthe

motorwillbeinsufficienttoremovethedamageifitsmaximumpossiblevalue

isw0 =0andkw0aαandβareknownconstants

337Onanairplane there isadevice(anaccelerometer) thatmeasures theaccelerationsnormaltotheaxisofthefuselageandintheplaneofthewingTheautomatic pilot is programed for a horizontal rectilinear flight with constantvelocity Because of errors in direction the angle Ψ(t) made by the velocityvectorwiththefixedverticalplaneisrandomEstimatetheaveragenumberoftimesperunit timeinwhichthesensitiveelementoftheaccelerometerwillgooffscale if thiseventoccurswhen the instantaneousradiusofcurvatureof thetrajectoryof theairplane in thehorizontalplanebecomesequal to theminimaladmitted radius of circulationR0 The velocity of the plane υ can be assumed

constantandwhereτ=t2ndasht1

338 The altitudeH(t) of an airplane directed by an automatic pilot is arandomfunctionwhoseexpectation isthegivenaltitudeofflightandwhose

correlationfunctionisAssuming that H(t) is normal find the minimal altitude that can be

establishedinthesystemofdevicesforpilotlessflightsothatduringtimeTtheprobabilityoffailurecausedbycollisionwiththesurfaceoftheearthislessthanδ=001percent ifa=400sqmα=001secndash1β=01 secndash1andT = 5hours

339 A radio control line insures the transmission of a signal withoutdistortioniftheperturbationX(t)attheinputofthereceiverduringtransmissiondoes not exceed in absolute value some level a Find the probability Q fortransmission without distortion if

andthetimeoftransmissionisT3310FindthedistributionlawfortheordinatesofanormalrandomX(t)at

itspointsofmaximaif3311GivenanormalstochasticprocessX(t)findthedistributionlawforthe

ordinatesofitsminimaif3312Estimate theaveragenumberof inflexionpointsofanormalrandom

functionX(t)intimeTif3313Estimate theaveragenumberofmaxima perunitareaofanormal

random function of two variables ζ(x y) if its two-dimensional correlationfunction is a function of two variables

anditstwo-dimensionalspectraldensity

isknown3314Under the assumptionsmade in the preceding problem estimate the

averagenumberofpoints perunitareainwhichbothfirstpartialderivatives

partζ(xy)partxandpartζ(xy)partychangetheirsignfromldquo+rdquotoldquondashrdquo

34 SPECTRALDECOMPOSITIONOFSTATIONARYRANDOMFUNCTIONS

BasicFormulasAnystationaryfunctionX(t)canbewrittenas

whereinthecaseinwhich

theincrementsdΦ(ω)satisfytherelations

HereSx(ω) is thespectraldensityof therandomfunctionX(t)andδ(x)denotestheδ-function(seeSection11page48)

ThecorrelationfunctionandspectraldensityarerelatedbymutuallyinverseFouriertransforms

whicharetheconsequenceofspectraldecompositionofX(t)Forτ=0thefirstoftheforegoingformulasleadsto

Thespectraldensitycannothavenegativeordinatesforrealfunctions

The random functionswith finite variance have spectral densities vanishing atinfinityfasterthan1ω

The spectral density of the derivative is related to Sx(ω) by the

formulaThe necessary and sufficient condition that a random function be (once)

differentiableis

whichholdsonlyifSx(ω)approacheszeroforincreasingωfasterthan1ω3If the random functions are stationary and stationarily correlated then

between themutual correlation functionRxy(τ) and themutual spectral density

Sxy(ω)thefollowingrelationsholdFromthedefinitionsofRxy(τ)andSxy(ω)itfollowsthat

The spectral density of the product of two normal (real) stationary randomfunctionsX(t)andY(t)

is expressed in terms of Sx(ω) Sy(ω) and Sxy(ω) by the formula

IntheparticularcasewhenY(t)equivX(t)Sy(ω)=Sxy(ω)=Sx(ω)wehaveZ(t)

=X2(t)andThesameresultcanbeobtainedbyusingaformulavalidforanytwonormal

(stationary)functions

andthenapplyingtheFouriertransformtoRxy(τ)


To solve Problems 341 to 3410 it is necessary to apply the FouriertransformdirectlyIndeterminingthecorrelationfunctionforthecaseinwhichthespectraldensityistheratioofpolynomialsinωtheusualwaytoobtaintheresult is by calculations To find the spectral density when one knows thecorrelation function and it involves the modulus of its argument the infinitedomainofintegrationmustbepartitionedintotwo(ndashinfin0)and(0infin)Intherestoftheproblemsitisnecessarytofindthecorrelationfunctionorspectraldensitybyusingtheirdefinitionsandinsomeproblemsalsobyusingthepropertiesofnormalvariables

Example341Findthecorrelationfunctionif

SOLUTIONUsingtheFouriertransformweget

For τ gt 0 is the integral of a functionofacomplexvariableωoveracontourformedbythereal

axisandaclosedsemicircleofinfiniteradiusintheupperhalf-planeThustheintegralrsquos value is calculated bymultiplying the residue of the function at theuniquepoleω=iλj(weconsiderReλjgt0)locatedinsidethecontourby2πie

πλjendashλjτandsoSimilarlyforτlt0byclosingtherealaxisthroughthelowerhalf-planewe

obtain that is for any sign of τ

Example342Findthespectraldensityif

SOLUTIONUsingthenotation

weseethat

Since

afterdifferentiationwithrespecttoαandsimpletransformationswefindthat

Example343Findthespectraldensity

ifX(t)isnormalrandomfunctionand

SOLUTIONSince

PROBLEMS

341Giventhespectraldensity

findthecorrelationfunctionK(τ)342Giventhespectraldensity

findthecorrelationfunctionK(τ)343FindthespectraldensityS(ω)if

344FindthespectraldensityS(ω)if


346 Find the spectral density



349AccordingtotheformofthespectraldensityofarandomfunctionX(t)determine how many derivatives this function has if

3410FindthespectraldensityS(a)if

3411 Find the values of the quotient αβ for which the spectral density

hasamaximumatω=03412FindthevarianceofthederivativeofarandomfunctionX(t)if

3413 Find the mutual spectral densities and if

3414ThecontrolsignalΔ(t)senttothecontrolunitsofanautomaticsystemisdefinedbytheformula

FindSΔ(ω)if

3415Adynamicalsystem(predictor)isusedtoobtainthevalueoftheinputrandomfunctionX(t)attimet+τ0whereτ0istheleadtimeofpredictionFindthemutualspectraldensitybetweenX(t)andY(t)=X(t+τ0)ifKx(τ)isknown

3416 A random functionX(t) is fed to the input of a dynamical systemFurthermore X(t) is the sum of a useful signal U(t) and noise V(t)

Theproblemofthedynamicalsystemisthecalculationofthefunction

FindthemutualspectraldensitySxy(ω)ifSv(ω)Su(ω)andSuv(ω)areknown3417FindthespectraldensitySz(ω)if

and ifX(t) andY(t) are independent random functionswith known correlationfunctions

3418FindthespectraldensitySz(ω)if

whereX(t)andY(t)areindependentrandomfunctionsKx(τ)=a1endashα1|τ|Ky(τ)=a2endashα2|τ|and and areknown

3419 The ldquoCardano errorrdquo Δ(t) which occurs by using a CardanosuspensioninsomeofthestabilitydevicesonshipsisrelatedtotheheelangleΘ(t)andtrimdifferenceangleΨ(t)bytheformula

Assuming that Θ(t) and Ψ(t) are independent random functions find thecorrelation function the variance and the spectral density of the errorΔ(t) if

and3420FindthespectraldensitySy(ω)if

whereX(t)isastationarynormalrandomfunctionand

3421FindthespectraldensitySy(ω)if

whereX(t)isanormalrandomfunction isknownand


whereX(t)isanormalrandomfunction

and isknown3423 The correction Δ(t) for the roll of a ship to the azimuth angle of

direction of a navigational radar station is defined by the formula

FindSΔ(ω) if q can be considered constant and the yaw angle Φ(t) trimdifference angle Ψ(t) and heel angle Θ(t) are uncorrelated normal randomfunctions with known correlation functions

3424 A normal random function X(t) has a correlation function

and expectation Find the maximum of the spectraldensitySy(τ)if

FIGURE34

3425Twoidenticaldiskswhoserotationaxescoinciderotatewithdifferent(incommensurable)angularvelocitiesΩ1andΩ2(Figure34)Inthesedisksthereare holes bounded by two radii making a central angle γ and by thecircumferencesofradiusrndash(12)Δandr+(12)ΔThecentersoftheseholesareselectedonthecircumferenceofradiusγaccordingtoauniformdistributionlaw

OnonesideofthedisksisapointsourceoflightLandontheothersideaphotocell F in front of which is placed a diaphragm D the aperture of thediaphragmhastheshapeofasectorwithangleГboundedbythecircumferencesof radius r ndash (12)Δ and r + (12)Δ The intensity of the photocurrent J isproportional to the sumof the areasof all theholeswithin theapertureof thediaphragmFindthespectraldensityfortheintensityofthecurrentSj(Δ)iftherearen holes in eachdisk and if it is equally probable that anyhole in the firstdiskindependentofthepositionsoftheotherholesislocatedoppositeaholeintheseconddiskatanyangulardistancefromtheopticalaxisofthesystemlightsourceandthephotoelement4(Neglectthecasewhenthesizeoftheapertureisdecreasedbythediaphragm)

35 COMPUTATIONOFPROBABILITYCHARACTERISTICSOFRANDOMFUNCTIONSAT

THEOUTPUTOFDYNAMICALSYSTEMS

BasicFormulasForanylineardifferentialequation

thegeneralsolutioncanberepresentedas

whereyj(t) is a systemof independentparticular integralsof thehomogeneousequationCj are constants determined by the initial conditions and they aregenerally speaking random quantities YI(t) is a particular integral of thenonhomogeneousequationanditsatisfieszeroinitialconditionsandisgivenby

theequalitywherep(tt1)istheGreenrsquosfunctionofthesystem(impulsefunction)expressedin terms of the particular integrals yj(t) by the formula

In the case in which the coefficients of the equation are constants theGreenrsquosfunctiondependsonlyonthedifferenceofthearguments

If the system is stable aj(t) = const and ifX(t) is stationary then for asufficientlylarget(comparedwiththetimeofthetransientprocess)thefunctionY(t) can also be considered stationary In this case

andKy(τ)canbefoundbyFourierinversionofSy(ω)IfX(t)isrelatedtothestationaryrandomfunctionZ(t)bytheformula

wehave

thelastformularemainingvalidevenwhenZ(t)doesnothaveanmthderivativehowevertheexpressionforSy(ω)decreasesfasterthan1ωwhenωincreases

IftheelapsedtimetfromthestartofoperationofthesystemisnotlargeifthefunctionX(t)isnonstationaryorifthecoefficientsoftheequationdependontimethentofindtheprobabilitycharacteristicsofthesolutionitisnecessarytoapply the general formulas for linear operators which (if for simplicity theconstants Cj and X(t) are uncorrelated) lead to

where||kjl||isthecorrelationmatrixofthesystemofrandomvariablesCjForequationswithconstantcoefficientswereplacep(t1t2)byp(t2ndash t1) in

thelastformulasIfX(t)isastationaryfunctionthen

wherey(ω t) isaparticular integralof theequationwithzero intialconditionsandwhereX(t)isreplacedbyeiωt

Inthiscase

A similar formula holds if X(t) is nonstationary but can be obtained bymultiplyingastationaryfunctionbyaknown(nonrandom)functionoftimeforexamplewhereX1(t)isstationaryInthiscasey(ωt)mustbelookeduponasaparticularintegral of the equation in which the right-hand side has been replaced by

b(t)eiωtieasbeforethestationaryfunctionhasbeenreplacedbyeiωtConsider a system of differential equations with constant coefficients

associatedwithastabledynamicalsystem

where ajl are constants Xj(t) are stationary random functions and time t issufficiently large Its solutionsare stationary randomfunctionswhose spectraldensitiesandmutualspectraldensitiescanbeexpressedintermsofthespectraldensitiesandmutualspectraldensitiesoftheright-handsidesoftheequationsas

followsHereΔ(ω)isthedeterminantformedfromthecoefficientsappearingonthe

left-handsidesoftheequations

whereAij(ω)isthecofactoroftheelementlocatedattheintersectionofthe ithrowandthejthcolumnandSxjxj(ω)equivSxj(ω)

The distribution law for the solution of a linear equation (systemof linearequations) whose right-hand side contains normal random functions andvariablesisalsonormalIftheequationislinearbutthedistributionlawoftherandomfunctionson the right-handside isnotnormal thedistribution lawforthesolutionalsowillnotbenormalTheexpectation andthecentralmomentsμj of this distribution law for any t are determined by the formulas

where X(t) is the random function appearing on the right-hand side of theequationand


Example 351 The error ε(t) in measuring the acceleration of an airplanewith the aid of an accelerometer is defined by the equation

whereγ(t)isarandomfunctioncharacterizingtherandomperturbationactingonthesensitiveelementoftheaccelerometerandSγ(ω)=c2asympconst

Find the variance of the velocity of the airplane by integrating theaccelerometer readings during timeT if no supplementary errors occur duringintegrationandthetimeforthetransientprocessismuchlessthanT

SOLUTION By assumption the error ε(t) can be considered a stationaryrandom function of time and thus

The error in velocity will not be stationary and its

variancewillbedefinedbytheformula Kε(t2ndasht1)dt1dt2Passing to thenewvariablesτ= t2ndash t1ξ= t2+ t1andcomputing the integralwith respect to ξ we obtain

Inasimilarwayonecansolvealltheproblemsinwhichtherequiredrandomfunctionisastationarysolutionofalinearequationwithconstantcoefficientsortheresultofapplicationofalinearoperatortoastationarysolution

Example352FortimetfindthevarianceofaparticularintegralY1(t)oftheequation [dY(t)dt] + aY(t) = tX(t) with zero initial conditions if

SOLUTIONInthisparticularcaseY(t)isnotstationarybecauseontheright-handsideoftheequationthereisanonstationaryfunctionoftime

Wehave

where

Since

then

and

whichafterintegrationleadsto

Example353Find the spectral density and themutual spectral density ofthe stationary solutions of the system of equations

if

SOLUTIONIfwereplacethedifferentialoperatorbyiωontheleft-handsidesthe determinant of the resulting system of algebraic equations becomes

Thecofactorsoftheelementsofthedeterminantare

Consequentlyapplyingthegeneralformulaweget

PROBLEMS

351 The input signal of a first-order dynamical system described by theequation

isarandomfunctionX(t)whosespectraldensityinthefrequencyband|ω|leω0

whereω0 αcanbeconsideredconstantFindthecorrelationfunctionofY(t)fort 1α352Adynamicalsystemisdescribedbytheequation

where =constisknownand a1a0gt0Findtheexpectationandvarianceforthestationarysolutionofthisequation353ThedeviationU(t)ofaheel-meter locatedintheplaneof themidship

frame is defined by the equation

where TheheelangleΘ(t)andthevelocityofthelateralshiftofthecenterofgravityoftheship asaresultoforbitalmotion can be considered uncorrelated random functions

andalltheconstantscontainedintheformulasareknownEvaluateSu(ω)354Anastaticgyroscopewithproportionalcorrectionislocatedonashipin

theplaneofthemidshipframeFindthevarianceforthedeviationαofitsaxisfromthedirectiongivenbythephysicalpendulumiftheangleαisdeterminedbytheequationAssumethetimeelapsedsincethestartofthegyroscopeissufficientlygreatsothatα(t) can be considered stationary determine the spectral densitySu(ω)byuse of the result of Proble 353 where

355 Find the spectral density and correlation function of the stationarysolution of the equation

ifX(t)hasthepropertiesofldquowhitenoiserdquothatisSx(ω)=c2=const356 The angular deviation Θ(t) of the coil of a galvanometer from the

equilibrium position in the case of open circuit is defined by the equation

whereIisthemomentofinertiaofthecoilristhefrictioncoefficientDistherigiditycoefficientofthethreadonwhichthecoilissuspendedandM(t) is theperturbing moment caused by the impact of molecules from the surroundingmedium

FindthespectraldensityandthecorrelationfunctionoftheangleΘ(t)ifthespectral density M(t) can be assumed constant and according to results ofstatisticalmechanics D=kTwherek isBoltzmannrsquos constant andT is theabsolutetemperatureofthemedium

357 Two random stationary functions Y(t) and X(t) are related by theequation

FindthespectraldensitySy(ω)forthestationarysolutionoftheequationifSx(ω)=[4π(ω2+1)]

358Doestheequation

containingon its right-handside thestationaryfunctionX(t)admitastationarysolution

359Findthevarianceoftheordinateofthecenterofgravityofashipξc(t)

onawavyseaifwheretheordinateofthewavefrontX(t)hasthecorrelationfunction

handω0 areconstantsdefinedby theparametersof theshipα isaparametercharacterizing the irregularityofwavesβ is thedominant frequencyofwavesandω0gehgt0

3510 The error given by an accelerometer measuring the horizontalacceleration of an airplane is defined by the equation

whereh=06secndash1n=628secndash1g=981msec2andtheheelangleγ(t)isastationary normal random function with a known correlation function

Find the variance of ε(t) for the stationary operating mode of theaccelerometer

3511ProveiftheinputsignalofalinearstabledynamicalsystemdescribedbyequationswithconstantcoefficientsisarandomfunctionX(t)withpropertiesofldquowhitenoiserdquo(Sx(ω)=c2)thenforasufficientlylongelapsedtimeafterthestartofoperationsthecorrelationfunctionoftheoutputsignalY(t)isdefinedby

theequalitywherep(t)istheGreenrsquosfunctionofthesystem

3512 Find the variance of the heel angle Θ(t) of a ship defined by theequation

ifthewaveslopeangleF(t)hasazeroexpectation

andtherollingprocesscanbeconsideredstationary3513AstationaryrandomfunctionY(t)isrelatedtothestationaryfunction

X(t) whose spectral density is known by the equation

wherekgehgt0

Find themutualspectraldensitySyx(ω)and themutualcorrelationfunctionRyx(τ)

3514Given

find the correlation functionY(t) for times exceeding the time of the transientprocess

3515 The input signal of a dynamical system with Greenrsquos function p(t)represents a stationary random function X(t) with zero expectation Find thevarianceofthedeviationoftheoutputsignalY(t)fromsomestationaryfunctionZ(t)ifKx(τ)andRxz(τ)areknown =0andthetransientprocessofthesystemcanbeconsideredfinished

3516UsingthespectraldecompositionofastationaryrandomfunctionX(t)find for time t 1a the variance for the integral of the equation

withzeroinitialconditionsif

3517Asaconsequenceoftherandomunbalanceofthegyro-motorplacedonaplatformwitharandomverticalaccelerationW(t) thedirectiongyroscope

precesseswithangularvelocityFindtheexpectationandvarianceoftheazimuthaldepartureα(t)attimetif

M[L]=0D[L]= Kw(τ)and areknownPHandgareknownconstantsandLandW(t)areuncorrelated

3518 Find the correlation function of the particular solution YI(t) of the

equationwithzeroinitialconditionsif

3519 Two random functions Y(t) and X(t) are related by the equation

FindKy(t1t2)ifKx(τ)=aendashα|τ|andiffort=0Y(t)=03520 Find the expectation and the correlation function of the particular

solutionoftheequation

withzeroinitialconditionsif =t

3521Findtheexpectationandthecorrelationfunctionofthesolutionofthe

differentialequation

if for t= t0ne0Y(t)=y0wherey0 is a nonrandomvariable and =1t

3522 Write the general expression for the expectation and correlationfunctionofthesolutionY(t)ofadifferentialequationofnthorderwhoseGreenrsquosfunctionisp(t1t2)ifontheright-handsideoftheequationtherandomfunction

X(t)appears andKx(t1t2)areknownandtheinitialvaluesofY(t)andthefirst(nndash1)derivativesarerandomvariablesuncorrelatedwiththeordinatesoftherandomfunctionX(t)withknownexpectationsejandwithcorrelationmatrix||kjl||(lj=12hellipn)

3523Giventhesystem

findthevarianceofY2(t)fort=05seciffort=0Y1(t)andY2(t)arerandomvariables uncorrelated to X(t) D[Y1(0)] = 1 D[Y2(0)] = 2

3524Findthevarianceforthesolutionsofthesystemofequations

fortimetiftheinitialconditionsarezeroand


fort=05secifSx(ω)=[2π(ω2+1)]andtheinitialconditionsarezero3526Theinputsignaltoanautomaticfrictionclutchservingasadifferential

rectifier is a random functionX(t) Find the variance for the rectified functionZ(t)andthevarianceoftherectifiedvelocityofitsvarianceY(t)iftheoperationof the friction clutch is described by the system of equations

where a andb are constant scale coefficients andKx(τ) = and thetransientprocessisfinished

3527Fort=1findthedistributionlawforthesolutionoftheequation

if for t = 0 Y(t) = Y0 and Y0 and X(t) are normal andmutually uncorrelated and

3528 The deviation U(t) from the vertical position of a plane physicalpendulumwhoseplaneofoscillationcoincideswiththediametralplaneofashipis defined by the equations

where all coefficients are constant and the yaw angleΦ(t) the angle of trimdifferenceΨ(t) theheelangleΘ(t)and thevelocitiesof thecoordinatesof the

center of gravity of the ship are normal stationaryuncorrelatedrandomfunctions

ExpressthespectraldensitiesSx(ω)Sy(ω)andSxy(ω)necessaryforfindingtheprobabilitycharacteristicsofU(t)onasimulatingsystemintermsofspectraldensitiesSφ(ω)Sψ(ω)Sθ(ω) and

3529Fortimet 1kfindtheasymmetrySkandexcessExofaparticular

solutionoftheequationwithzeroinitialconditionsifX(t)isanormalstationaryfunction =0Kx(τ)=aendashα|τ|

3530FindthemutualcorrelationfunctionRyz(τ)ofthestationarysolutions

oftheequationswheretherandomfunctionX(t)hasthepropertiesofldquowhitenoiserdquo(Sx(ω)asympc2)k1gth1gt0k2gth2gt0

36 OPTIMALDYNAMICALSYSTEMS

BasicFormulasBy an optimal dynamical system5 we mean a system that for an input

functionX(t)=U(t)+V(t)whereU(t)istheusefulsignalandV(t)isthenoisehasanoutputfunctionY(t)whoseexpectationisequaltotheexpectationofsomefunctionZ(t)andThefunctionZ(t)isrelatedtotheusefulsignalU(t)by

whereNisaknownoperatorandn(tt1)isitsGreenrsquosfunctionTo find an optimal system is to determine according to the probability

propertiesoftherandomfunctionsU(t)andV(t)andtheformoftheoperatorNtheformof theoperatorLor itscorrespondingGreenrsquos function l(t t1)so thatthe function X(t) can be transformed into the function Y(t)

Theproblemofdeterminationofanoptimaldynamicalsystemcanbesolvedifthefollowinghold

(a) the random functions U(t) and V(t) are stationary and stationarilyconnectedandNandLarelinearoperatorsindependentoftime(b)thespectraldensitySx(ω)=Su(ω)+Sv(ω)+Suv(ω)+ is a rational functionof its

argumentItcanbeexpressedaswhere the polynomialsPm(ω) andQn(ω) have roots located only in the upperhalf-plane of the complex variable ie they can be represented as

wherethecomplexnumbersandμjandvlhavepositiveimaginarypartsmjandnl are the multiplicities of the corresponding roots

(c)inthedeterminationoftheordinatesofthefunctionY(t)onecanusethevaluesoftheordinatesofthefunctionX(t)foraninfinitelylongtimeprevioustothe current time t In this case the transmission functionL(iω) of the optimaldynamical system related to the Greenrsquos function by

isdefinedinthefollowingway(weassumethat =0)Ifthesystemoperateswithoutdelay(thatisZ(t)istheresultofapplication

ofsomeoperatortothepresentorfuturevaluesoftheordinatesofthefunction

U(t)then

where

and λr (r = 1 2 hellip α) is the pole of multiplicity lr (of the expression

locatedintheupperhalf-planeIftheoptimaldynamicalsystemmustoperatewithdelay(thatisthefunction

Z(t)istheresultofapplicationofsomeoperatortotheordinatesofthefunctionU(t) at an instant preceding the present time t by τ0 seconds) then

where

and κr (r = 1 2 hellip αprime) is the pole of multiplicity of the expression

locatedinthelowerhalf-planeThevarianceD[ε(t)]fortheoptimaldynamicalsystemis

If thedynamicalsystemmakesuseof theordinatesof therandomfunctionduringa finite intervalof time (tndashT t) preceding the present time t (ldquosystemwithfinitememoryrdquo)andtheusefulsignalisthesumofthepolynomialRk(t)ofa preassigned degree k (the coefficients of the polynomial being arbitraryconstants)andastationaryrandomfunctionU(t)thatistheinputfunctionX(t)is

thenunder the sameassumptionsabout the formof the spectraldensitySx(ω)the Greenrsquos function l(τ) of the optimal dynamical system is defined by theformulas

Hereαraretherootsoftheequation|Pm(iα)|2=0N(iω)isthetransmissionfunctionoftheoperatorNandtheconstantsontherightsideoftheequalityaredetermined by substituting the expression for l(τ) in the equation

satisfiedbytheGreenrsquosfunctionl(τ)oftheoptimaldynamicalsystemandthenequating the coefficients of equal powers in t as well as those of equal

exponential functions To the 2n + k + 1 equations thus obtained should beaddedthek+1equationsformedbyequatingthemomentsofthefunction l(τ)andtheGreenrsquosfunctionn(τ)associatedwiththeoperatorNietheequations

where

Thesystemofequations thusobtainedcompletelydefinesall theconstantscontained in the expression for l(τ) The transmission function L(iω) can be

foundfroml(τ)byaFouriertransformandthevarianceoferrorε(t)fortheoptimalsysteminthepresentcaseis

InasimilarwayonecansolvetheproblemoffindingtheGreenrsquosfunctionofanoptimaldynamicalsystemifthenonrandompartoftheusefulsignalcontainsalinearcombination(withconstantbutunknownparameters)oftrigonometricorexponential functionsof timeTheonlydifference is that in theexpression forl(τ) a similar linear combination will appear whose coefficients can bedeterminedbysubstitutionintheinitialintegralequation

In some problems one prefers not to form optimal dynamical systemsbecauseofdifficultiesconnectedwiththeirpracticalrealizationandinsteadoneformssystemsthatarenotoptimalinthestrictmeaningofthewordbutthatgivetheminimalvarianceD[ε(t)]amongsystemswhoserealizationintheparticularcase presents no special difficulties For example to find the value of the

functionU(t)attimet+τonecantakeasY(t)anddeterminea1anda2sothatfor

For such a statement of the problem the determination of the form ofoperator L (the values of the constants appearing in the expression for this

operator)reducestothedeterminationoftheextremumofafunctionofseveralvariables


Example 361 A dynamical system is designed to give the bestapproximationoftherandomfunctionZ(t)=NU(t+τ0)Findthemutualspectraldensity Sxz(ω) if X(t) = U(t) + V(t) and the transmission function N(iω) ofoperatorN the prediction time τ0 the spectral densities Su(ω)Sv(ω) and themutualspectraldensitySuv(ω)areknown

SOLUTIONSettingU+V(insteadofX(t))intheexpression

replacingU(t)andV(t)bytheirspectraldecompositionsandtakingintoaccount

that after simple transformations weobtain


Example362TherandomfunctionX(t)=U(t)+V(t)isfedintotheinputofa dynamical system where the spectral density of the useful signal Su(ω) =α2(ω2 + β2)Suv(ω) = 0 and the spectral density of noise can be consideredconstant Sv(ω) = c2 Find the transmission function L(iω) of the optimaldynamicalsystemifthejobofthesystemistoproducethefunctionZ(t)=U(t+τ)where(a)τge0(b)τlt0

SOLUTIONInthiscase

(a)Forτge0theexpression hasonepoleintheupper half-plane ω = iβ consequently

(b) For τ lt 0 has one pole in the lower half-plane ω = ndash iγ consequently

Example363ThedistanceD(t) toanairplanemeasuredwith theaidofaradardevicewitherrorV(t)istheinputtoadynamicalsystemthatestimatesthepresentvalueofthevelocitybytakingintoaccountonlyitsvaluesduringtime(tndashTt)DeterminetheoptimalGreenrsquosfunctionl(τ)if thecorrect value of the distance can be quite accurately approximated by apolynomialofthirddegreeintσv=30mα=05secndash1β=20secndash1andT=20sec

SOLUTION Since to the correlation function Kv(τ) there corresponds the

spectral density and the useful part of therandomsignalU(t)=0theninthenotationsassumedinthisexamplewehavek = 3 n ndashm = 1 Sx(ω) = Sv(ω) the numerator of Sv(ω) contains noω andconsequentlyithasnoroots

Greenrsquosfunctionoftheoptimalsystemwillbe

Todeterminetheconstantsaftersubstitutingl(τ)intheequation

weequatethecoefficientsofequalexponentialfunctions

Adding to theseequations theequalitiesobtainedbyequating themomentsof l(τ) and n(τ) = δ(1)(τ)

weobtainacompletesystemof linearequationswhichdeterminetherequiredconstantsSolvingthissystemwefind

PROBLEMS

361Attheoutputofadynamicalsystem

emergeswhereU(t)isausefulsignalandV(t)isthenoiseFindSx(ω)ifSu(ω)Sv(ω)andSuv(ω)areknown

362 At the output of a dynamical system designed to receive a function

afunctionX(t)=U(t)+V(t)emergeswhereV(t)denotesthenoiseadded in the receptionof theordinatesof functionU(t)Find themutualspectraldensitySxz(ω)ifSu(ω)Suv(ω)andSv(ω)areknown

363Find the transmission functionL(iω) of an optimal dynamical systemdesignedtoreceivethederivativeoftherandomfunctionX(t)duringτseconds

beforethelastobservationoftheordinateofX(t)ifFindthevarianceoftheerrorintheestimateofthevelocity364FindthetransmissionfunctionL(iω)ofanoptimaldifferentiablesystem

ifthesystemservestodeterminethederivativeofarandomfunctionU(t)attimetndashτ(τgt0)andifattheoutputthesignalisarandomfunctionX(t)thatisthesum of a useful signalU(t) and noise V(t) not related toU(t) Assume that

365Findthe transmissionfunctionofanoptimalfilterdesignedtoreceivethepresentvalueofausefulsignalifitsinputsignalconsistsofthesumoftheuseful signal U(t) and the noise signal V(t) U(t) and V(t) are mutually

uncorrelatedand366 Express the variance of the error of an optimal dynamical system in

terms of the spectral densitiesSu(ω)Sv(ω) andSuv(ω) (U(t) denotes a usefulsignalandV(t) thenoise) if thetransmissionfunctionof theoptimalsystemisL(iω)andNistheoperatorthatappliedtothefunctionU(t)minimizestheerrorinthesystem

367Attheoutputofadynamicalsystemdesignedtoreceivethederivative

X(t)+U(t)+V(t)emergeswhere thenoiseV(t)and thesignalU(t)areuncorrelated

Findtheoptimaltransmissionfunctionofthesystemandthevarianceofthe

errorintheestimateofthederivative 368Findtheoptimaltransmissionfunctionofadynamicalsystemdesigned

toreceivethevaluesoftheordinateofU(t+τ)iftheinputsignalisrepresented

byarandomfunctionU(t)369ThespectraldensityoftheinputsignalisSx(ω)=1(ω+1)2andτge0is

the prediction time Find the optimal transmission function of the dynamicalsystem

3610Thespectraldensityoftheinputsignalis

Find the optimal transmission function of a dynamical system designed toproduceX(t+τ)andthevarianceoftheerrorintheestimateofX(t+τ)forτge0

3611 The input to a dynamical system consists of the sum of twouncorrelatedfunctionsusefulsignalU(t)andnoiseV(t)Determinetheoptimaltransmission function for the evaluation of the signal at time t + τ if τ ge 0

3612 The input to a delay filter consists of the sum of two uncorrelatedfunctions signalU(t) and noiseV(t) whose correlation functions are known

Findtheoptimaltransmissionfunctionofthedynamicalsystemandtheerrorinfilteringifthedelayisτ0(τ0ge0)

3613Thespectraldensityof the inputsignal isSx(ω)=α2(ω4+4α4)andthe prediction time is τ (τ ge 0) Find the optimal transmission function of thedynamicalsystemdesignedforthedeterminationofX(t+τ)

3614Onarollingshipitisnecessarytodetermineatimetsothatτ0secondslater the linear function of the heel angle Θ(t) and its derivative n1 Θ(t) +

(wheren1andn2areknownconstants)willassumeaprescribedvaluecFindtheoptimaltransmissionfunctionofthepredictorandthevariance of

theerrorif =03615Thecoordinateofashipmovingonarectilinearcoursewithaconstant

velocityisestimatedwithanerrorV(t)characterizedbythecorrelationfunction

whereσv=25mandα=025secndash1Findthemaximalaccuracyattainedinestimatingthevelocityofvariationof

thecoordinatefortheobservationtimesT=2040and240seconds3616 Under the assumptions of the preceding problem find themaximal

accuracy attained in the estimate of the velocity of variation of the shipcoordinateif

andalltheotherconditionsarethesame

3617Toestimatethepresentvaluesoftheangularrollingvelocity ofashiponeusesadynamicalsystemtheinputtothissystemisthepresentvalueof the heel angle Θ(t) distorted by an error of measurement V(t) Find thevarianceoftheerrorε(t)intheestimateoftheangularvelocityifthissystemcanbeconsideredoptimalwith =0 Rθv(τ)equiv0

[cosβτ+ (αβ)sinβ|τ|]σθ=01 radα=01secndash1β =075secndash1σv=210ndash2radandαv=05secndash1

3618Adynamical systemhasbeendesigned to determine thevalues of arandomfunctionX(t)attimet+τ0accordingtothevaluesoftheordinatesofthisfunctionduringtheinterval(tndashTt)FindtheoptimaltransmissionfunctionofthesystemandthevarianceoftheerrorinthedeterminationofX(t+τ0) if themeasurementsoftheordinatesoffunctionX(t)areperformedpracticallywithouterrorswherec1 and c2 are unknown constants andU(t) is a random functionwhosecorrelation function is

3619AdynamicalsystemobtainsthederivativeofarandomfunctionX(t)attime t + τ0 Find the optimal transmission function of the system if

wherec1andc2 areunknownconstants and the systemhas a ldquofinitememoryrdquo(thatisusesonlythevaluesofX(t)duringtheinterval(tndashTT))σu=1α=01secndash1τ0=10secandT=40sec

3620 Find theGreenrsquos function l(τ) of an optimal dynamical systemwithldquofinitememoryrdquoTdesignedforthedifferentiationofthefunctionX(t)=R1(t)+

U(t)andfindtheerrorinthedeterminationof whereR1isapolynomial

offirstdegreeand3621 For automatic control of airplanes one can use an inertial control

systemconsistingofdevicesoftwotypesinthefirstcaseduringtheoperationof the system the following signal is determined

wherec1c2c3c4aresome(unknown)constantsandΩ=125middot10ndash2secndash1 in

the second case the signal has the form

Find theoptimal transmission functionsof thedynamical systemsused forthe determination of the signal in both cases if the systems have a ldquofinitememoryrdquoTT=20secandtheusefulinputsignalisdistortedbyanerrorV(t)

3622ThepredictingvalueoftherandomfunctionX(t+τ0)isY(t)=aX(t)Findthevalueof theconstanta thatminimizes thevarianceof theerrorε(t)=aX(t) ndash X(t + τ0) and the minimal value of the variance if = 0

3623 The predicting value of the random function X(t + τ) is the linear

combinationZ(t)=aX(t) +b Find the values of constants a and b thatminimize the variance of the error

andtheminimalvarianceofthiserrorif =0

3624Thepredictingvalueof therandomfunctionU(t+τ0) isY(t)=a[U(t)+V(t)]whereV(t) is the error in the estimate of the present value of the usefulsignalU(t) Find the value of the constant a that minimizes the variance of

if

3625Asignalmustbesenttopredictthezerovalueofthederivativebyτ0 secondsActually the signal is sentat the instant inwhich the following

linearcombinationbecomeszeroFind the optimal values of constantsab and c and the magnitude of the

variance of (t + τ0) if = 0

σθ=5degβ=07secndash1α=0042secndash1andτ0=02sec3626 Under the assumptions made in the preceding problem find the

optimal values of the constants a b and c for which

37 THEMETHODOFENVELOPES

BasicFormulasAny normal stationary function X(t) can be represented for = 0 as

wheretherandomfunctionsA(t)andΦ(t)aremutuallyuncorrelatedThefunctionsX(t)andY(t)=A(t)sinΦ(t)haveamutualcorrelationfunction

that can be expressed in terms of Sx(ω) by the relation

whereRxy(τ)vanishesforτ=0ConsequentlyforequaltimesthefunctionsX(t)andY(t)areuncorrelatedandbeingnormaltheyalsoareindependent

The distribution laws for the ordinates of the functions A(t) and Φ(t) areuniquely defined by the correlation function according tothefollowingformulastheone-dimensionaldistributiondensities

thetwo-dimensionaldistributiondensities

wherea1φ1 and a2φ2 are the values for the amplitude and the phase of the

envelopeattimestandt+τq2=1ndashk2(τ)ndashr2(τ)κ=κ(τ)= cos(φ2ndashφ1ndashγ)γ=γ(τ)=arctan[r(τ)k(τ)]andI0(z)istheBesselfunctionofthefirstkindofzeroorderandofanimaginaryargument

Theprecedingformulasleadtotheconditionaldistributionlaws

andtheformulaforthecorrelationfunction

whereK(k2)andE(k2)denotethetotalellipticintegralsoffirstandsecondkinds

The four-dimensional and two-dimensional distribution laws for theamplitudeof theenvelope itsphaseand thecorrespondingvelocitieshave theform

where

Theprobabilitythat isgreaterthanzeroisdefinedby

Similarly

For a narrow-band spectrum of the random variable X(t) the quantity is small compared to and some of the foregoing

formulas can be simplified by expanding the corresponding expressions inpowersofthesmallquotientΔω1Inparticularforanarrow-bandspectrumthe

variances and become small and since M[A(t)] = 0M[Φ(t)]=ω1 bydifferentiating the random functionX(t)=A(t) cosΦ(t)one

mayconsiderinsomecasesthat vanishesandreplace byω1Inthecaseofanarrow-bandspectrumtheprobabilitydensityofthetimeτ

during which the random function is above (below) the zero level (ldquothedistribution lawof thehalf-periodrdquo)has the followingapproximateexpression

whoseaccuracyincreaseswiththedecreaseofthequotientΔω1


Example 371 Find the average number of passages per unit time for the

randomfunction

whereΦ(t)isthephaseofthenormalrandomfunctionX(t)if

SOLUTIONWedeterminethespectraldensity

Consequently

Applyingthegeneralformulafor thenumberofpassagesperunit timeweobtain

SinceΘ(t)=Φ(t)ndashω1tΘhasauniformdistributionlawintheinterval(0

2π)andthedistributionlaw canbeobtainedeasilyifwereplace by + ω1 in the distribution law that is

where

Setting intheformulaforpweget

PROBLEMS

371Thecorrelationfunctionisdefinedbytheformula

Considering X(t) normal ( = 0) find the correlation function for theamplitudeoftheenvelopeofthisfunction

372 What is the probability that the phase of the envelope of a normalrandom function X(t) will decrease if

373ForastationarynormalrandomfunctionX(t)findtheprobabilitythatthe phase will increase (decrease) if

374FindtheprobabilityPthatthevelocityofvariationofthephaseofthe

envelopewillbegreaterthan

if

375 For a normal random functionX(t) find the distribution law for thevelocity of variation of the phase if

376FindthedistributionlawforthephaseofanormalrandomfunctionX(t)ndash forwhich

377Findthedistributionlawforthevelocityofphasevariationofanormalrandom function X(t) with spec tral density

378Findthedistributionlawfortheenvelopeandthevelocityofvariationof the envelope of a normal random function X(t) if

379 Under the assumptions made in the preceding problem find theconditionaldistributionlawoftheenvelopeattimet+τifattimet

3710 Find an approximate expression for the distribution law of the timeduring which a random function is below the zero level if

3711Assumingthattheformulasfortheenvelopeofarandomfunctionwithanarrow-bandspectrumareapplicablefindthedistributionlawfortheintervalsbetween successivemoments duringwhich the deck of a ship passes throughequilibriumiftheheelangleΘ(t)isanormalrandomfunctionwhosecorrelationfunction

andthereisnopitching3712 Find the average number of passages beyond the level 2σx per unit

time for a random functionA(t) ifA(t) is the envelope of the normal randomfunctionX(t)and

3713 Find the average number of passages beyond the level 2σx for theamplitude of the envelope of a normal stochastic process X(t) if

3714FindtheconditionaldistributionlawforthephaseofanormalfunctionX(t) at time t + τ if at time t the phase is zero and

Neglecting the variance of the amplitude of the envelope determine thevariance of X(t) at time (t + πω1 where

3715Findthemutualcorrelationfunctionfortwonormalstationaryrandomfunctions X(t) and Y(t) if

1WhennototherwisespecifiedX(t)isreal2X(t)isconsideredreal3 For a more rigorous definition of the notion of ldquooperatorrdquo see Taylor A E Introduction to

FunctionalAnalysisNewYork JohnWileyampSons Inc 1958 andHeiderL J andSimpson JE TheoreticalAnalysisPhiladelphiaWBSaundersCompany1967

4SuchadevicewasproposedbyVSGytelrsquoson5ThereareotherpossibledefinitionsofthenotionofanoptimaldynamicalsystemForexampleby

optimal system one can understand a system forwhich the probability that the differenceY(t) ndashZ(t) inabsolute value does not exceed a prescribed quantity is maximal The term ldquodynamical systemrdquo isunderstood in the technical sense of theword ie itmeans any systemwhose state (characterized by afunction obtained at its output) changes because of the influence of external perturbations (randomfunctionsattheldquoinputrdquoofthesystem)


38 MARKOVCHAINS

BasicFormulasLetS be a finite sample space consisting of outcomesQ1Q2hellipQm A

sequenceoftrialsoftheunderlyingexperimentiscalledafiniteMarkovchainifpij(k)theconditionalprobabilityatthekthtrialofQjundertheassumptionthatQioccurredatthe(kndashl)sttrialisindependentoftheoutcomesatthe(kndash2)nd(kndash3)rdhelliptrialsTheeventsQ1Q2hellipQm arecalledstatesof theMarkovchainandthekthtrialcanbeconsideredasthechangeofstateattimetk

In each column of matrix there is at least one elementdifferentfromzeroandthetransitionprobabilitiesPij(K)(ij=12hellipm)for

anyksatisfytherelationAMarkov chain is called irreducible if any state canbe reached fromany

otherstateandperiodicifthereturntoanystatecanbemadethroughanumberofstepswhichareamultipleofsomeκgt1

AMarkovchain is calledhomogeneous if the transitionprobabilitiesPij(k)areindependentofkthatisPij(k)=pij(ij=12hellipm)

The column p(n) = p1(n)p2(n)hellippm(n) which is formed of theunconditionalprobabilitiesthatatthenthtrialthesystemwillpassrespectivelyto states Q1 Q2hellip Qm is defined by the formula

andforahomogeneouschainby

where the accent means transposed matrix that is if then

Foranynbutrelativelysmallmtocalculate wecanusetheLagrange-Sylvesterformulawhichinthecaseofsimpleeigenvaluesλ1λ2hellipλm(rootsof

the equation where is the unit matrix) has the form

Inthegeneralcaseforfinding it isconvenient toreduce tonormalform =HJHndash1where j is a diagonal or a quasidiagonalmatrix dependingonlyon theeigenvaluesofmatrix For simpleeigenvalues whereδik=0forinekandδkk=1TheelementsofmatricesHandHndash1arethesolutions of algebraic equations of the form H = HJ Hndash1 = JHndash1

Then where for simple eigenvalues

Theelements ofmatrix arealsodeterminedbythePerronformula

where r is the number of distinct eigenvalues vs is their multiplicity

and Aji(λ) is the cofactor of the element λδji ndash pji in the

determinant Thematrix ofthelimitingtransitionprobabilities and the column p(infin) = ( infin)p(0) of the limiting unconditionalprobabilities canbeobtained from thecorrespondingexpressionbypassage tothelimitfornrarrinfinThelimitsexistonlyif |λs|lt1fors=23hellipr (for thetransition probability matrices |λs| le 1 always obtains and one eigenvalue λ1equals unity) For this

wherev1isthemultiplicityoftheeigenvalueλ1=1For v1 = 1 allm rows of matrix are equal and the elements of the

column p(infin) coincide with the corresponding elements of any row that is

Inthiscasetheprobabilities canalsobedeterminedfromthesolutionof

the algebraic system

If the finiteMarkov chain is irreducible and nonperiodic then to find theprobabilities onecanusethelastequationsIfthenumberofstatesm=infinthe Markov chain is irreducible and nonperiodic and the system of linearequations has a nontrivial solution forwhich and probabilities

are the solutions of the system(j=12hellip)where

Ifonecanseparateagroupofstatesofthesystemsothatatransitionfromanystateofthisgrouptoanyoftheremainingstatesisimpossiblethegroupcanbeconsideredan independentMarkovchainAgroupmayconsistofonestateQksothatpkk=1Qkiscalledanabsorbingstate

InthegeneralcasefromthestatesQ1Q2hellipQmonecanselectmutuallydisjointgroupsC1C2hellipChcalledessentialstates therest formagroupTofinessentialstatesForapropernumberingofstatesthematrix isreducedto

theformwhereR1R2hellipRharethematricesoftransitionprobabilitiesofthegroupsC1C2hellipChWisasquarematrixassociatedwiththeinessentialstatesofgroupTandUisanonzero(ifthereareinessentialstates)notnecessarilysquarematrix

IfalltheeigenvaluesofmatricesR1R2hellipRhexceptthoseequaltounityare less than unity in absolute value then

whereUinfinissomerectangularmatrixLeth=1inthematrix ie thereisonegroupCofabsorbingstatesIf

the Markov chain formed from the states of this group is nonperiodic theprobabilities pj of transition from an inessential state Qj to the group C of

essentialstatesisdeterminedfromtheequationwhereinthefirsttermthesummationisextendedoverinessentialstatesandinthesecondovertheessentialstates

Let κj (j = 1 2hellip h) be the number of eigenvalues (considering theirmultiplicity) of thematrixRj that are not exactly equal to unity but equal inmodulustounityTheminimalcommonmultiplicityoftheseeigenvaluesistheperiodκoftheMarkovchainIfthechainisirreducibleallstatesoftheperiodicchaincanbedivided intogroupsG0G1 hellipGκndash1 so that a transition from astatecontainedinGralwaysleadsinonesteptoastateinGr+1(Gκ=Go)IntheMarkovchainwithmatrix κeachgroupGrcanbeconsideredan independentchainthe following limits for r = 0 1 hellip κndash1 exists

theprobabilitiespkκaredeterminedasinthecaseκ=0In the general case there also exists a matrix and matrices

The matrix ofmean limiting transition probabilities is defined by the formula

The column ofmean limiting unconditional probabilities is given by

If h = 1 in the matrix then the mean limiting unconditional

probabilities (j = 1 2 hellip m) are uniquely defined by the equalities


Example381Somenumbersareselectedatrandomfromatableofrandomnumbers containing integers 1 tom inclusive The system is in stateQj if the

largestoftheselectednumbersisj(j=12hellipm)Findtheprobabilities(ik = 1 2hellipm) that after selectingn random numbers from this table thelargestnumberwillbekifbeforeitwasi

SOLUTION Any integer 1 to m appears equally probable in the table ofrandom numbers and thus any transition from stateQ1 (the largest selectednumberis1)toanystateQjisequallyprobableThenp1j=1m(j=12hellipm)The transition fromQ2 toQ1 is impossible and consequently p21 = 0 ThesystemcanremaininstateQ2intwocasesiftheselectednumberis1or2andconsequentlyp22=2mp2j=1m(j=34hellipm)Inthegeneralcasewefind

Thematrixoftransitionprobabilitiescanbewrittenas

Thecharacteristicequation

hasrootsλk=km(k=12hellipm)Tofindtheprobabilities representingthe elements of the matrix let us apply Perronrsquos formula The cofactors

Aki(λ) of the elements of the determinant are the following

SubstitutingtheseexpressionsinPerronrsquosformulaweobtain

InasimilarwayonecansolveProblems383to3810

Example382AvendingmachinethatsellstokensinasubwaystationcanbeoperatedwithnickelsanddimesIfanickelisinsertedthemachinereleasesone token if thecontainerwhichcanholdmnickels isnot fullotherwise themachinereleasesnotokenIfadimeisinsertedthemachinereleasesonetokenand a nickel change if there is at least one nickel in the container if not themachine turns off One knows that a nickel and a dime are inserted with

probabilitiespandq=pndash1Findtheprobabilities (ik=01hellipm)thatafterndemandsfortokensthemachinewillcontainknickelsifinitiallyitheldinickels

SOLUTIONLetthestateQjmeanthatthecontainerhasjnickels(j=01hellipm)Forllejlemndash1atransitionfromQjtoQj+1ispossiblewithprobabilitypand to Qj ndash 1 with probability q When the states Q0 or Qm representingabsorbing states are reached the machine turns off Therefore

Thematrixoftransitionprobabilitieshastheform

whereW is a squarematrix of orderm ndash 1 andU andV are two columns ofordermndash1

wherethematrixWisassociatedwiththeinessentialstatesQlQ2hellipQmndash1Therequiredprobabilitiesaretheelementsofthematrix

andconsequently

TofindtheelementsofmatrixWnformthecharacteristicequationΔmndash1=|λ ndashW|=0Fordeterminantsofthisty|pethereobtainsthefollowingrecursionrelationwithΔ0=1Δ1=λThen

Thelasttermoftheequationis foroddmandforevenm

Making the substitution we can write theequationΔmndash1=0intheform

Fromthisitfollowsthatμk=expi(kπm)(k=12hellipmndash1)Thereforetheeigenvalues will be

The matrix W can be reduced to the form W = HJHndash 1 where J =

andH=||hjk||istobedeterminedThematrixequationWH=HJisequivalenttothefollowingequations

Uptoafactorthesolutionsofthissystemaretheelements

Thus TheinversematrixHndash1canbewrittenintheform

FromHHndash1= wefindCk=2m(k=12hellipmndash1)Usingtheequalitywn=HJnH ndash 1 we obtain

Todetermine theelements (j=12hellipm ndash1)of thecolumnUnweshallusePerronrsquosformulaThecharacteristicpolynomialof thematrix will

be ForthecofactorsofA0j(λ)oftheelementsofthedeterminant|λ ndash |wegetthefollowingexpressions

Then

wheretheasteriskmeansthatthefactorwithk=vmustbeeliminatedfromtheproduct

Theprobabilities (j=1 2hellipm ndash 1) can be calculated similarlyToevaluate them we can also use the equalities

Problems3811to3814maybesolvedsimilarly

Example383Asubstanceisirradiatedbyastreamofradioactiveelementsduring equal time intervals Δt The probability that during irradiation thesubstancewillabsorbr radioactiveparticles isdeterminedby the formulaβr=arrendashaEachradioactiveparticlecontainedinthesubstancemaydecayduringtwosuccessiveirradiationswithprobabilityqFindthelimitingprobabilitiesforthenumberofparticlesinthesubstance

SOLUTION Let state Qi mean that after an irradiation the substance willcontaini(i=01hellip)radioactiveparticlesDuringtheintervalΔtthetransitionfromQitoQkwilloccurifindashvparticles(v=01hellipi)decayandkndashv(kgev)are absorbed by the substance The transition probabilities are

wherep=1ndashqandsummationisextendeduptoiifilekanduptokifkltiThesubstancecancontainanynumberofparticlesieallthestatesofthe

system are attainable Therefore the Markov chain is irreducible Sinceprobabilitiespiiaredifferentfromzerothechainisnonperiodic

Letusconsiderthesystemofequations

Weset

andmultiplyboth sides of the systemby zj sumover j from0 toinfin and thenapply the formula n ndash 1 times Hence

Fromthiswefindthat

ComparingthetwoexpressionsforG(z)weobtain

Since and the arbitrary constant G(l) can be takendifferent from zero and infinity the algebraic systemhas a nontrivial solutionandtheseries isconvergentConsequently canbedeterminedfrom the system The system for

is similar to the preceding system solved for uj and therefore

Since G(1)=1andthustherequiredprobabilitiesare

OnecansolveProblems3816to3822inasimilarway

Example384ThenumberXofdefectiveitemsineachindependentsampleofsizeNselectedfromaninfinitelylargelotobeysabinomialdistributionlawthat isP(X=k)=pk = (k=0 1hellipN)q = 1 ndashp If a samplecontains r defective items then according to the acceptance criteria oneconsiders the lot as changing its preceding stateQv toQv + r ndash 1 The lot isrejectedifv+rndashlgemandacceptedifv+rndash1=0FindtheprobabilitythatthelotwillbeacceptedifitsinitialstateisQj(j=12hellipmndash1)

SOLUTIONTherearem+1statesQi(i=01hellipmndash1)possibleIfthestateQ0isreachedthelotisacceptedifQmisreacheditisrejectedSincethesetwoareabsorbingstatesp00=1pmm=1Ifine0andinemPii+jndash1=Pj(j=01

hellipmndashi)pim=1ndash (i=12hellipmndash1)Thematrixoftransitionprobabilitiesis

The required probabilitiespj (j = 1 2hellipm ndash 1) are the probabilities oftransitionfrominessentialstatesQ1Q2hellipQmndash1totheessentialstateQ0andcan be determined from the algebraic system

whichcanbewrittenintheform

ThedeterminantΔmndash1ofthissystemcanbefoundbytherecursionformula

whereΔ0=1Therequiredprobabilitiesaredeterminedbytheequations


Example 385 A truck transports goods among 2m points located on acircular route These goods are carried only from one point to the next withprobability p or to the preceding point with probability q = 1 ndash p Find theprobabilities (jk=12hellip2m)thataftern transports thetruckwillpassfromtheythpointtothekthpointEvaluate theseprobabilitiesfornrarrinfinandcomputethemeanlimitingprobabilitiesoftransition

SOLUTION Let stateQj (j = 1 2hellip 2m)mean that the truck is at the kthpoint The transition probabilities are

Thematrixoftransitionprobabilitiesis

LetusintroducethematrixH=||hjk||=||ε(jndash1)(kndash1))||oforder2minwhichε=eπim By direct multiplication we find that

and consequently the eigenvaluesof willbe (k=12hellip2m)

Theeigenvalueswithmaximalabsolutevalueareλ1 = 1 andλm + 1=ndash1theyhavemultiplicityoneandthusthechainisperiodicwithperiodκ=2The

inversematrixFrom the equality = HJnH ndash 1 where Jn = || λk δjk || we find

whichcanbewrittenas

Alltermsinthesumexceptthefirstaresmallerthanunityinmodulussothatfornrarrinfin

Thisimpliesthat

The last equalities can bewrittenwithout using the expression for as anirreduciblechainandthetransitioninonestepfromthegroupC0ofstateswithodd numbers always leads to the groupC1 of states with even numbers andconversely

Themeanlimitingtransitionprobabilitiesare

UsingthissolutiononecansolveProblems3826and3827

Example386IndiscussingthefundamentalstatementsofkinetictheoryofmatterEhrenfestproposedthefollowingmodelmmoleculesdistributedintwocontainers are randomly removed one by one fromone container to the other

Findthemeanlimitingunconditionalprobabilitiesforthenumberofmoleculesinthefirstcontainer

SOLUTION Let the state Qi mean that there are i molecules in the firstcontainer(i=01hellipm)Thenplindash1=imPii+1=1ndashim(i=01hellipm)The matrix of transition probabilities can be written as follows

FromanystateQiareturntoQiispossibleonlyinanumberofstepsthatisamultipleof twoTherefore in thepresentcase theMarkovchain isperiodicwithperiodκ = 2The chain is irreducible because each state can be reachedfromanyotherstate

The column of mean limiting unconditional probabilities can be

determined from the condition that is

Fromthisitfollowsthat Usingtheequality wefind that consequently therequiredprobabilities

areSimilarlyonecansolveProblems3828and3829

PROBLEMS

381ShowthatforahomogeneousMarkovchainthetransitionprobabilities are correlated by the equality

382 Given the column of initial probabilities p(0) = α β γ) and thematrices of transition probabilities for times tl t2 t3

determinethecolumnofunconditionalprobabilitiesp(3)383Accordingtotherulesofacompetitionacontestantquitsamatchifhe

loses twopoints inonegameor if thereare twotiesAcontestantwithout tiescanwinateachgamewithprobabilityαcantiewithprobabilityβandcanlosewithprobability1ndashαndashβIncaseofonetietheprobabilityofwinningateachgameisγFind theprobabilityof losingvariousnumbersofpoints inngamesforthecontestantwhoseoutcomesinthepreviousgamesareknown

384 If thecurrent inanelectriccircuit increases theblockingsystemofacertain device fails with probability α and the entire device ceases to operatewith probability β If the blocking system fails then at the next increase ofcurrentthedeviceceasestooperatewithprobabilityγFindtheprobabilitiesthatno failurewill occur in the circuit that only theblocking systemwill fail andthatthedevicewillceasetooperateafternincreasesincurrentiftheinitialstateofthedeviceisknown

385 There are several teams in a certain competition During each roundonlythreemembersofateamcancompetewithanotherteamAccordingtotherules of the competition no ties can occur and the one who loses once iseliminatedfromthiscompetitionLetαβandγbetheprobabilitiesthatinthenextroundinturnamongonetwoandthreemembersremainingrespectivelyfromateamnonelosesletβ1andγ1betheprobabilitiesthatinthenextroundin turnamongtwoandthreeremainingteammembers respectivelyone losesand let γ2 be theprobability that two among threemembers of this team lose

Determinetheprobabilities (ik=0123)thatafternroundskmembersof this team compete if before these rounds i members of the same teamcompeted

386Anautomaticsystemcanoperate if fromN identicalunitsmndash1 faileach unit can fail only during an operation cycle The probabilities pik of

transitionof the systemduringonecycle fromstateQi to stateQk are knownwheretheindexofastaterepresentsthenumberofunitsthatfailedsothatforkltipik=0(ik=01hellipm)pmm=1Prove that the transitionprobabilities

for n cycles during which the defective units are not replaced withprobabilities Pk = Pkk (k = 0 1 hellip m) are determined by the formulas

forigtk (k=01hellipm)andforkgti

where

387Provethatifundertheassumptionsmadeintheprecedingproblempkk= p (k = 0 1 hellip m ndash 1) then


whereDki(λ)isdeterminedbytheformulaoftheprecedingproblemforpk=p(k=01hellipmndash1)

388 From an urn containingN white and black ballsm balls are drawnsimultaneously The black balls are used to replace the white balls that aredrawn Initially the urn containsm white balls and after several drawings itcontains iwhiteballsDetermine theprobabilities (ik=0 1hellipm) thatafternadditionaldrawingstherewillbekwhiteballsintheurnEvaluatetheseprobabilitiesforN=6m=3

389Foragivenseriesofshotseachmarksmanfromonegroupscoresanynumber of points ranging from N + 1 to N + m with equal probabilitiesDeterminetheprobabilitythatamongthenextnmarksmenofthisgroupatleastone will scoreN + k points if the maximal number of points scored by thepreviousmarksmenisN+i(kgei=12hellipm)

3810 Along a straight line AB in a horizontal plane there are placedidentical vertical cyclinders of radius r whose centers are a distance l apart

Perpendicular to this line spheres of radius R are thrown and the path of amovingspherecrossesABwithequalprobabilityatanypointoftheintervalLonwhichtherestandmcylindersThedistancebetweenthecentersofthecylindersis l gt 2(r + R) each time a sphere hits a cylinder the number of cylindersdecreasesbyoneDeterminetheprobabilities (ik=01hellipm)thatafternthrowskcylinderswillremainifbeforethistherewereicylinders

3811 In a domain D partitioned into m equal parts points are placedsuccessivelysothattheirpositionsareequallyprobablethroughoutthedomainDetermine the probabilities (i k = 1 2hellipm) that after placing a newseries ofn points the number of parts ofD containing at least one pointwillincreasefromitok

3812Attimes tl t2 t3hellipashipcanchange itsdirectionbyselectingoneoutofmpossiblecoursesQ1Q2hellipQmTheprobabilitypijthatattimetrtheshipchangesfromQitoQjispij=αmndashi+j+1andαm+k=αkne0(k=12hellip

m) Determine theprobability that for tn lt t lt tn + l thedirectionoftheshipwillbeQkiftheinitialdirectionwasQj(jk=12hellipm)Findthisprobabilityforn=infin

3813 Consider the following model of the diffusion process with centralforceAparticlecanlieonlyonthesegmentABatpointswithcoordinatesxk=xA+kΔ(k=01hellipm)wherexm=xB It shifts stepwise fromxj to thenextpoint toward A with probability jm and to the next point toward B withprobability1ndashjmDeterminetheprobabilities (ik=01hellipm)thatafternstepstheparticlewillbeatpointxkifinitiallyitwasatxi

3814 The assumptions here are the same as in Example 382 but themachinedoesnotturnoffWhentherearenonickelsinthecontainerandadimeisinsertedortherearemnickelsandanickelisinsertedthemachinereturnsthelastcoininsertedwithoutreleasingatokenFindtheprobabilities (ik=01hellipm)thatafterndemandsfortokenstherewillbeknickelsinthecontainerifinitiallytherewereinickels

3815TwomarksmenAandBfireshotsinturnsothataftereachhitAfiresand after each failureB firesThe right for the first shot is determinedon thesame basis by reference to the outcome of a preliminary shot fired by arandomlychosenmarksmanDeterminetheprobabilityoffailureatthenthtrialindependent of the previous hits if the probabilities of failure at each trial forthesetwomarksmenareαandβrespectively

3816 Given the matrix of transition probabilities that isirreduciblenonperiodicand twice-stochastic ie the sumofelementsofeachcolumnandofeachrowisunityfindthelimitingprobabilities (j=12hellipm)

3817Therearemwhite andm blackballs that aremixed thoroughlyandthenequallydistributedintwournsFromeachurnoneballisrandomlydrawnandplacedintheotherFindtheprobabilitiespik(ik=01hellipm)thatafteraninfinitenumberof such interchanges the firsturnwill containkwhiteballs ifinitiallyitcontainediwhiteballs

3818AsegmentABisdividedintomequalintervalsAparticlecanlieonlyonthemidpointofsomeintervalandshiftsstepwisebyanamountequaltothelengthofoneintervaltowardpointBwithprobabilitypandtowardpointAwithprobabilityq=1ndashpAt theendpointsofAB reflecting screens areplaced sothatuponreachingAorBtheparticleisreflectedtowarditsinitialpositionFindthelimitingunconditionalprobabilities (k=12hellipm)thattheparticleisineachofthemintervals

3819GiventhefollowingtransitionprobabilitiesforaMarkovchainwithaninfinite number of states

determinethelimitingprobabilities (j=12hellip)3820ThetransitionprobabilitiesforaMarkovchainwithaninfinitenumber

ofstatesisdefinedbypi1=qpii+1=p=1ndashq(i=12hellip)Findthelimiting

probabilities (j=12hellip)3821AMarkov chainwith an infinite number of states has the following

transition probabilities

Findthelimitingprobabilities (ik=12hellip)3822Aparticlemakesarandomwalkonthepositiveportionofthex-axis

Theparticle can shift byone stepΔ to the rightwithprobabilityα to the leftwith probability β or it can remain fixed it can reach only points withcoordinatesxj(J=12hellip)Fromthepointwithcoordinatex1=Δtheparticlecanmovetotherightwithprobabilityαorremainfixedwithprobability1ndashαFindthelimitingtransitionprobabilities (k=12hellip)

3823Thematrixoftransitionprobabilitiesisgivenintheform

whereR is thematrix associatedwith the irreducible nonperiodic groupC ofessentialstatesQ1Q2hellipQs and the squarematrixW is associatedwith theinessentialstatesQs+1Qs+2hellipQmDeterminethelimitingprobabilitiespj(j=s+1s+2hellipm)thatthesystemwillpassintoastatebelongingtogroupC


whereR is the matrix corresponding to the nonperiodic groupC of essentialstatesQ1Q2hellipQs and the squarematrixW corresponds to the inessentialstatesQr+lQr+2hellipQmFindtheprobabilitiesPj(j=r+1r+2hellipm)thatthesystemwillpassintoastatebelongingtothegroupCifalltheelementsofWareequaltoαandthesumofelementsofanyrowofmatrixUisβ

3825TwoplayersAandBcontinueagameuntilthecompletefinancialruinofoneTheirprobabilitiesofwinningateachplayarerespectivelypandq(p+q=1)Ateachplaythewinofoneplayer(lossfortheother)isonedollarandthe total capital of the players is m dollars Determine the probabilities offinancial ruin foreach ifAhas jdollars (j=1 2hellipm ndash1)before thegamebegins

3826Giventhetransitionprobabilitiespjj+1=1(j=12hellipmndash1)pm1=

1 determine the transition probabilities and themean limiting transitionprobabilities

3827Thematrixoftransitionprobabilitiesis

whereαne1Determine the transitionprobabilities and themean limitingtransitionprobabilities (jk=1234)

3828Giventheelementsofthematrixoftrasitionsprobabilities

withoutevaluating theeigenvaluesof thematrix find the limiting transitionprobabilitiesandthemeanlimitingunconditionalprobabilities

3829AparticleisdisplacedonasegmentABbyrandomimpactsandcanbeatthepointswithcoordinatesxj=xA+jΔ(j=01hellipm)Reflectingscreensare placed at the endpointsA andB Each impact can shift the particle to therightwithprobabilitypandtotheleftwithprobabilityq=1ndashpIftheparticleisnext to a screen any impact shifts it to the screen in questionFind themeanlimitingunconditionalprobabilities that theparticle isateachdivisionpointofthesegmentAB

39 THEMARKOVPROCESSESWITHADISCRETENUMBEROFSTATES

BasicFormulasThebehaviorofasystemwithpossiblestatesQ0Q1Q2QmcanbedescribedbyarandomfunctionX(t)assumingthevaluekifattimetthesystemisinstateQk If the passage fromone state to another is possible at any time t and theprobabilitiesPik(tτ)oftransitionfromstateattimettostateQkattimeτ(τget)are independentof thebehaviorof the systembefore the time t thenX(t) is aMarkov stochastic process with a discrete number of states (The number ofstates can be finite or infinite) The transitionprobabilitiesPik(t τ) satisfy the

relation

Theprocessishomogeneousif

InthiscasefortheMarkovprocess

AMarkovprocessiscalledregularif(a)foreachstateQkthereexistsalimit

(b) for each pair of states Qi and Qk there exists a temporal transitionprobability density pik(t) continuous in t defined by

wherethelimitexistsuniformlywithrespecttotandforfixedkuniformlywithrespecttoi

For regularMarkovprocesses theprobabilitiesPik (tτ) are determined bytwosystemsofdifferentialequations

withinitialconditions

where

ForahomogeneousMarkovprocessci(t)andPij(t)areindependentoftimePik(t τ) = Pik(τ ndash t) and the systems of differential equations become

withintialconditions

TheprobabilitiesPk(t) that the system is in stateQk at time t is given by thesystem of equations

withcorrespondinginitialconditionsforPj(t)IftheinitialstateQiisgiventhe

initialconditionsareForhomogeneousMarkovprocessesthelastsystembecomes

andtheinitialconditionsare

IfforahomogeneousMarkovprocessthereexistsatimeintervaltgt0suchthatPik(t)gt0forall iandk then theprocess iscalled transitiveandfor it the

limitexistsindependentoftheindexoftheinitialstateThelimitingprobabilitiespkinthis case are determined from the system of algebraic equations

The equations for probabilities Pik(t τ) and Pi(t) can be obtained either byapplyingtheforegoinggeneralformulasorfindingthevariationsofprobabilitiesfordifferentstatesofthesystemduringasmalltimeintervalΔtandpassingtothelimitasΔtrarr0

An example of a Markov process is the simple flow of events with thefollowingproperties

stationarity that is foranyΔtgt0and integerkge0 theprobability thatkeventswilloccurduringtheinterval(tt+Δt)isthesameforalltge0absenceofaftereffectthatistheprobabilityofoccurrenceofkeventsduringtheinterval(t t + Δt) is independent of the number of occurrences before the time tordinaritythatis

whereR2(Δt)istheprobabilitythatatleasttwoeventsoccurduringintervalΔt


Example391A system can be in one of the statesQ0QQ2hellip and itpassesduringtimeΔtintoastatewhoseindexishigherbyonewithprobabilityλΔt+o(Δt)FindtheprobabilitiesPik(t)oftransitionfromstateQitostateQk(kgei)duringtimet

SOLUTIONTheprocessisMarkovianbyassumptionMoreoveritisregularsince

andotherwisepik=0Consequently the equations for homogeneous Markov processes are

applicable

with initial conditions Pik(0) = δik Multiplying both sides of the obtainedequations by uk and summing over k from i to infin we get

where Thesolutionofthelastequationhastheform

Sincebydefinition

wehave

ComparingthelastexpressionwiththedefinitionofG(tw)weobtain

Theinitialsystemofdifferentialequationsforpik(t)canalsobeobtainedinanotherway theprobabilityPik(t+Δt) is thesumof theprobabilityPik(t)[1 ndashλΔtndasho(Δt)] that thepassage from stateQi to stateQk (k gt i) occurred duringtimeTandtheprobabilityPi kndash1(t)[λΔt+o(Δt)] that thispassageoccurs in theinterval (t t + Δt) that is

TransposingPik(t)totheleftsideoftheequalitydividingbothsidesbyΔtandpassingtothelimitasΔtrarr0weobtaintherequiredequationInthismannertheequationfork=icanbededuced

Problem396mayabesolvedinasimilarway

Example 392 A queuing system consists of a large (practically infinite)numberof identicaldevices eachdevice servicingonlyonecall at a timeandspending on it a random time that obeys an exponential distribution lawwithprobability densityμendashμt The incoming calls for service form a simple queuewith parameter λ Evaluate (a) the probability Pn(t) that at time t exactly ndeviceswill be busyn lem) if initially all deviceswere free (b) the limitingprobabilitiespn=limtrarrinfinPn(t)(c)theexpectednumberofdevicesbusyattimet

SOLUTIONSincethequeueofcallsissimpleandtheservicingtimeobeysanexponential distribution during the time interval (t t + Δt) the system willchangeitsstatemorethanoncewithaprobabilitywhoseorderofmagnitudeishigherthanΔt

ThereforeconsideringonlythefirstndashordertermsduringtimeintervalΔtweobtain

Thesystemisregularbecause

(a)We substitute the calculated values for cnpn n + 1 andPn n ndash1 in thesystem of differential equations for Pn(t)

fornge1and

Ifoneassumesthatattimet=0alldevicesarefreetheinitialconditionsare

Theresultingsystemcanbesolvedwiththeaidofthegeneratingfunction

Multiplyingbothsidesofdifferentialequationsbyunandsummingaftersimpletransformationswefind

TheinitialconditionisG(0u)=1Theresultinglinearnonhomogeneouspartialdifferentialequationisreplaced

byanequivalenthomogeneousone1

withinitialconditionV=Gndash1fort=0Tosolvethelastequationitisnecessaryfirsttosolvethesystemofordinary

differentialequations

whoseindependentintegralsare

Usingtheinitialconditionst=0u=u0G=G0weobtaintheCauchyintegrals

ofthesystemThe right-hand sides are the principal solutions of the homogeneous partialdifferential equationUsing these solutionswe form the solution ofCauchyrsquosproblem for the homogeneous partial differential equation

ThesolutionoftheCauchyproblemfortheinitialequationisthefunctionGforwhichV=0hence

The probabilities Pn(t) are related to the generating function G(t u) by theequality

whichleadsto

thatisaPoissonlawwithparameter

(b)Thelimitingprobabilitiespnareobtainedfromtheinitialonesbypassagetothelimit

that ispn obey a Poisson distribution lawwith parametera =λμ (The sameresult can be obtained if we solve the system of algebraic equations obtainedfromthedifferentialsystemforPn(t)afterreplacingPn(t)bypnand[dPn(t)dt]byzero)(c)Theexpectednumberofbusydevicesis

ForM(t)writethedifferentialequation

Sinceinitiallyalldevicesarefree

Problems3917to3919maybesolvedinasimilarway

Example393AqueuingsystemconsistsofmdeviceseachofwhichatanygiventimecanserviceonlyonecallItservicesforarandomtimeobeyinganexponentialdistributionlawwithparameterμTheincomingcallsformasimplequeuewithparameterλAcallisservicedimmediatelyafteritisreceivedifthereisatleastonefreedeviceatthattimeotherwisethecallisrejectedanddoesnotreturntothesystemDeterminethelimitingprobabilityforarejectedcall

SOLUTIONLetQidenotea stateof the system inwhich idevicesarebusythenPik(t)gt0forafinitetimeintervalConsequentlywecanapplyMarkovrsquostheorem stating that there exist limiting probabilities such that

anddeterminedbytheformula

Asintheprecedingexamplewehave

and the other probabilities pjk = 0 Substituting these values for pjk in theequations for pn we get

Ifwesetzn=λPnndash1ndashnμPnthesystembecomes

henceitfollowsthatzn=0forallnandthismeansthat

ThesystemiscertainlyinoneofstatesQn(n=012hellipm)therefore

fromthistheprobabilityp0thatalldevicesarefreeis

Theprobabilitythattheserviceisrefusedis

Following this solution Problems 398 3910 3911 and 3914 may besolved

PROBLEMS

391 The particles emitted by a radioactive substance in the disintegrationprocess forma simple flowwithparameterλEachparticle can independentlyreachacounterwithprobabilitypDeterminetheprobabilitythatduringtimetnparticleswillberecordedbythecounter

392 Two communication channels feed two independent simple flows oftelegrams to a given point Find the probability that n telegrams will arriveduringtimetiftheparametersofthecomponentflowsareλ1andλ2

393TheelectronicemissionofthecathodeofanelectronictuberepresentsasimpleflowofelectronswithparameterλTheflighttimesfordifferentelectronsare independent random variables with the same distribution function F(x)Determinetheprobabilitythatattime tafter thestartofemissiontherewillbeexactlynelectronsbetweentheelectrodesofthetubeanddeterminethelimitingprobabilityofthesameevent

394 For a simple flow of events determine the correlation coefficientbetweenthenumberofoccurrencesintheintervals(0t)and(0t+τ)

395ForarandomtimeTnofoccurrenceof thenthevent inasimpleflowwith parameter λ determine the distribution function Fn(t) the probabilitydensityfn(t)andthemomentsmk

396FindthetransitionprobabilitiesofasystemfromstateQttostateQkintime t in a homogeneous Markov process if in a single change of state the

systemcanpassonlyfromstateQn tostateQn + 1and theprobability that thesystemwillchangeitsstateduringtimeinterval(tt+Δt)is[λΔt+o(Δt)]

397ThecustomersofarepairshopformasimplequeuewithparameterλEachcustomerisservicedbyonerepairmanduringarandomtimeobeyinganexponentialdistributionlawwithparameterμIftherearenofreerepairmenthecustomer leaveswithout serviceHowmany repairmenshould therebe so thatthe probability that a customer will be refused immediate service is at most0015ifμ=λ

398 One repairman servicesm automatic machines which need no careduring normal operation The failures of each machine form an independentsimple flow with parameter λ To remove the defects a repairman spends arandomtimedistributedaccordingtoanexponentiallawwithparameterμFindthe limitingprobabilities thatkmachinesdonot run (arebeing repairedorarewaitingforrepairs)andtheexpectednumberofmachineswaitingforrepairs

399SolveProblem398undertheassumptionthatthenumberofrepairmenisr(rltm)

3910AcomputeruseseitherunitsoftypeAorunitsoftypeBThefailuresoftheseunitsformasimpleflowwithparametersλA=01unitshourandλB=001unitshourThetotalcostofallunitsoftypeAisaandthatofallunitsoftype B is b (b gt a) A defective unit causes a random delay obeying anexponential distribution lawwith an average time of two hours The cost perhourofdelayiscFind theexpectationfor thesavingachievedbyusingmorereliableelementsduring1000hoursofuse

3911 The incoming calls for service in a system consisting of nhomogeneousdevicesformasimplequeuewithparameterλTheservicestartsimmediately if there isat leastone freedevice andeachcall requiresa singlefreedevicewhose servicing time is a randomvariableobeying an exponentialdistributionwithparameterμ(μngtλ)Ifacallfindsnofreedevice itwaits inline

Determine the limiting values for (a) the probabilities pk that there areexactly k calls in the system (being serviced and waiting in line) (b) theprobabilitypthatalldevicesarebusy(c)thedistributionfunctionF(t)andtheexpectedtimetspentbyadevicewaitinginline(d)theexpectednumberm1ofcallswaitinginlinetheexpectednumberm2ofcallsintheservicingsystemandtheexpectednumberofworkingdevicesm3thatneednoservice

3912Themachinesarrivingat a repair shop thatgivesguaranteed serviceforma simplequeuewithparameterλ=10unitshourThe servicing time forone unit is a random variable obeying an exponential distribution law with

parameter μ = 5 unitshour Determine the average time elapsed from themomentamachinearrivesuntil it is repaired if thereare four repairmeneachservicingonlyonemachineatatime

3913Howmany positions should an experimental station have so that anaverage of one per cent of itemswaitmore than 23 of a shift to start if theduration of the experiments is a random variable obeying an exponentialdistributionlawwithameanshiftof02andtheincomingdevicesusedintheseexperimentsformasimplequeuewithanaveragenumberof10unitspershift

3914Aservicingsystemconsistsofndeviceseachservicingonlyonecallat a time The servicing time is an exponentially distributed random variablewith parameter μ The incoming calls for service form a simple queue withparameterλ(μngtλ)AcallisservicedimmediatelyifatleastonedeviceisfreeIfalldevicesarebusyandthenumberofcallsinthewaitinglineislessthanmthecallslineupinthewaitinglineiftherearemcallsinthewaitinglineanewcallisrefusedservice

Findthelimitingvaluesfor(a)theprobabilitiespkthattherewillbeexactlykcallsintheservicingsystem(b)theprobabilitythatacallwillbedeniedservice(c) theprobabilities that all servicingdeviceswillbebusy (d) thedistributionfunctionF(t) for the timespent in thewaiting line (e) theexpectednumberofcallsm1 in thewaiting line the expected number of callsm2 in the servicingsystemandtheexpectednumberofdevicesm3freedfromservice

3915AbarbershophasthreebarbersEachbarberspendsanaverageof10minutes with each customer The customers form a simple queue with anaverage of 12 customers per hour The customers stand in line if when theyarrive there are fewer than three persons in the waiting line otherwise theyleave

Determine the probability po for no customers the probability p that acustomer will leave without having his hair cut the probability p that allbarbers will be busy working the average number of customers m1 in thewaiting line and the average number of customers m2 in the barbershop ingeneral

3916 An electric circuit supplies electric energy tom identical machineswhichneedserviceindependentlyTheprobabilitythatduringtheinterval(tt+Δt)amachinestopsusingelectricenergyisμΔt+o(Δt)andtheprobabilitythatit will need energy during the same interval is [λΔt + o(Δt)] Determine thelimitingprobabilitythattherewillbenmachinesconnectedinthecircuit

3917A shower of cosmic particles is caused by one particle reaching theatmosphereatsomegivenmomentDeterminetheprobabilitythatattimetafter

thefirstparticlereachestheatmospheretherewillbenparticlesifeachparticleduringthetimeinterval(tt+Δt)canproducewithprobability [λtΔ+o(Δt)]anewparticlewithpracticallythesamereproductionprobability

3918AshowerofcosmicparticlesisproducedbyoneparticlereachingtheatmosphereatsomegivenmomentEstimatetheprobability thatat time tafterthefirstparticlereachestheatmospheretherewillbenparticlesifeachparticleduring the time interval (t t+Δt) canproduceanewparticlewithprobability[λΔt+o(Δt)]ordisappearwithprobability[μΔt+o(Δt)]

3919 In a homogeneous process of pure birth (birth without death) anumberofnparticlesattimetcanchangeinton+1particlesduringtheinterval

(tt+Δt)withprobabilityλn(t)Δt+o(Δt)whereor theycan fail to increase innumberDetermine theprobability thatat time ttherewillbeexactlynparticles

40 CONTINUOUSMARKOVPROCESSES

BasicFormulasA continuous stochastic process U(t) is called a Markov process if the

distribution function F(un | u1 hellip unndash1) of the ordinate of U(t) at time tncomputedundertheassumptionthatthevaluesoftheordinatesu1u2hellipunndash1attimestlt2helliptnndash1areknown(t1ltt2lthelliplttnndash1lttn)dependsonlyonthevalue

ofthelastordinateieTheconditionalprobabilitydensityf(un|unndash1)isafunctionf(txτy)offour

variables where for the sake of brevity one uses the notations

Thefunctionf(txτy)satisfiestheKolmogorovequations2

where

Thefunctionf(txτy)hasthegeneralpropertiesoftheprobabilitydensity

andsatisfiestheinitialcondition

Iftherangefortheordinatesoftherandomfunctionisboundedthatis

theninadditiontothepreviouslymentionedconditionsthefunction

shouldalsobeconstrainedbythefollowingboundaryconditions

(G(τy)mayberegardedasaldquoprobabilityflowrdquo)AsetofnrandomfunctionsU1(t)hellipUn(t) formsaMarkovprocess if the

probability density (distribution function) f for the ordinatesY1Y2hellipYn ofthese functions at time τ calculated under the assumption that at time t theordinates of the random functions assumed the values X1 X2 hellip Xn isindependent of the values of the ordinates ofU1(t)U2(t)hellipUn(t) for timesprevioustotInthiscasethefunctionfsatisfiesthesystemofmultidimensionalKolmogorov equations

wherethecoefficientsajandbjlaredeterminedbytheequations

andtheinitialconditions

Given the differential equation for the components of a Markov processU1(t)U2(t)hellipUn(t) to determine the coefficientsaj and bjl (a and b in thelinear case) onemust compute the ratio of the increments of the ordinates ofUj(t)duringasmalltimeintervalto(τndasht)findtheconditionalexpectationsoftheseincrementsandoftheirproductsandpasstothelimitasτrarrt

ToanymultidimensionalKolmogorovequationtherecorrespondsasystemofdifferentialequationsforthecomponentsoftheprocess

where ξm(t) are mutually independent random functions with independentordinates (ldquowhitenoiserdquo)whosecorrelation functionsareKm(τ)=δ(τ)and thefunction ψl and glm are uniquely determined by the system

TosolvetheKolmogorovequationsonecanusethegeneralmethodsofthetheoryofparabolicdifferentialequations(see forexampleKoshlyakovGlinerandSmirnov1964)WhenalandblmarelinearfunctionsoftheordinatesUl(t)thesolutioncanbeobtainedbypassingfromtheprobabilitydensity f(txlhellipxn τ yl hellip yn) to the characteristic function

obeying a partial differential equation of first order which can be solved by

generalmethods1Ifthecoefficientsalblmareindependentoftthentheproblemoffindingthe

stationary solutions of the Kolmogorov equations makes sense To find thestationary solutionof the secondKolmogorovequation setdfdτ=0and lookfor thesolutionof theresultingequationasafunctionofy1y2hellipynonly Intheparticularcaseofaone-dimensionalMarkovprocessthesolutionisobtainedbyquadratures

Any stationary normal process with a rational spectral density can beconsideredasacomponentofamultidimensionalMarkovianprocess

TheprobabilityW(T)thattheordinateofaone-dimensionalMarkovprocessduringatimeT=τndashtafteratimetwillwithknownprobabilitydensityf0(x)fortheordinatesoftherandomfunctionremainwithinthelimitsoftheinterval(α

β)iswheretheprobabilitydensityw(τy) is thesolutionof thesecondKolmogorovequationwithconditions

Whentheinitialvalueoftheordinateisknownf0(y)=δ(yndashx)Theprobabilitydensity f(T) of the sojourn time of a random function in the interval (α β) is

definedbytheequalityThe average sojourn time of the random function in the interval (α β) is

relatedtow(τy)by Forαneinfinβ=infinthelastformulasgive the probabilityW(T) of sojourn time above a given level the probabilitydensityf(T)ofthepassagetimeandtheaveragepassagetimeT

Theaveragenumberofpassagesbeyondthelevelαperunittimeforaone-dimensionalMarkovprocess is infinityHowever theaveragenumbern(τ0)ofpassagesperunittimeforpassageswithdurationgreaterthanτ0gt0isfiniteandfor a stationary process it is defined by the formula

wheref(α)istheprobabilitydensityfortheordinate(correspondingtoargumentα)oftheprocessandv(τy)isthesolutionofthesecondKolmogorovequationfor a stochastic process with conditions

which is equivalent to the solution of the equation for the Laplace-Carsontransform (p y) For a stationary process

Thetransformofn(τo)is

The probability W(T) that the ordinate U1(t) of a component of amultidimensional Markov process will remain within the interval (α β) ifinitiallythedistributionlawforthecomponentsU1(t)U2(t)hellipUn(t)isknownis defined by the equation

wherew(τ y1 hellip yn) is the probability density that the components of theprocessreachavolumeelementdy1hellipdynattimeτundertheassumptionthatduringtheinterval(tτ)theordinateU1(t)hasneverleftthelimitsoftheinterval(αβ) The functionw(τylhellip yn) is the solution of the secondKolmogorovequation with the conditions

Theprobabilitydensityf(T)ofthesojourntimeofU1(t)intheinterval(αβ)is defined by the formula

In the last formula α can be ndashinfin or β can be +infin which correspond toprobabilitiesofsojourntimeneitherabovenorbelowagivenlevel


Example 401 Write the Kolmogorov equations for a multidimensionalMarkovprocesswhosecomponentsU1(t)U2(t)hellipUn(t)satisfythesystemofdifferential equations

whereψj areknowncontinuous functionscj areknownconstants andξj(t)areindependent random functions with the property of ldquowhite noiserdquothat is

SOLUTIONTowrite theKolmogorovequations it suffices todetermine thecoefficientsajandbjloftheseequations

DenotingbyXjtheordinateoftherandomfunctionUj(t)attimetandbyYjits ordinate at time τ and integrating the initial equations we obtain

Consideringthedifferenceτndashtsmallwecancarryψjoutsidethefirst integralwithaprecisionuptosecondordertermsandsett1=TU1=XlU2=X2hellipUn= Xn which leads to

thatis

AssumingthattherandomvariablesX1hellipXnareequaltox1hellipxnfindingtheexpectation of the last equality and passing to the limit as τrarr t we obtain

Multiplying the expression for (Yj ndashXj) by that for (Yl ndashXl) and finding theexpectation of the product obtained we get

whichafterdivisionby(τndasht)andpassagetothelimitgives

Example 402 Given the first Kolmogorov equation for the conditionalprobability density f(t xl x2 τ y1 y2) of a normal Markov process

determinethesystemofdifferentialequationssatisfiedbythecomponentsU1(t)andU2(t)

SOLUTIONAccordingtothenotationsforthecoefficientsoftheKolmogorovequationswehave

Therequiredsystemofequationshastheform

whereξm(t)istheldquowhitenoiserdquowithzeroexpectationandunitvarianceBythegeneral formula given in the introductory section p 258 we have

Consequently

andtherequiredsystemhastheform

EliminatingU2(t) from the last equation we obtain forU1(t) a second-order

equation

Example403AnormalstationaryprocessU(t)hasthespectraldensity

where

and αj and βj are known constants Considering U(t) as a component of amultidimensionalMarkovprocessdeterminethecoefficientsoftheKolmogorovequationsofthisprocess

SOLUTION A stationary normal random function with rational spectraldensityisthesolutionofalineardifferentialequationcontainingldquowhitenoiserdquoon the right side In the present case the equation has the form

Weturnfromthenthorderequationcontainingthederivativesofξ(t)onitsrightsidetoasystemofequationsoffirstorderwithoutderivativesofξ(t)ontherightsideLetU(t)=U1(t) and introducenewvariablesdefinedby theequalities

where cl are arbitrary condstants for the time being The foregoing equationsformasystemofnndash1equationsoffirstorderTogetthelast(nth)equationintheinitialnth-orderdifferentialequationitisnecessarytoexpressallderivativesofUintermsofUjandtheirfirstderivativesPerformingthesetransformationswe obtain

Determining the coefficients cj so that the derivatives of ξ(t) disappear in theequation we find the recursion relations

whichforthelastequationofthesystemgives

Sincethecomponentsofann-dimensionalprocesssatisfyasystemoffirstorderequationsontherightsidesofwhichthereistheldquowhitenoiserdquotheprocessisann-dimensionalMarkov process The coefficients of theKolmogorov equationsaredeterminedasinExample401

Example404Theconditionalprobabilitydensity f(tx1x2τy1y2) of atwo-dimensional stochastic process U1(t) U2(t) satisfies the equation

whereαandβareconstantsDeterminethesystemofdifferentialequationssatisfiedbyU1(t)andU2(t)

SOLUTIONThegiven equation represents the secondKolmogorov equationandconsequentlytheprocessisatwo-dimensionalMarkovprocess

Thecoefficientsoftheequationare


whereξ1(t)andξ2(t)areuncorrelatedrandomfunctionsoftheldquowhitenoiserdquotypewithunitvarianceAccordingtothegeneraltheorytodetermineglmoneshould

solve the algebraic system of equations

Henceitfollowsthat

Consequentlytherequiredsystemhastheform

Example 405 Determine the asymmetry Sk and the excess Ex of theordinate of a random function Z(t) defined by the equality

ifζ(t)isanormalrandomfunctionζ=0Kζ(τ)=σ2endashα|τ|andthetransientphaseoftheprocessisassumedtohaveended(comparewithProblem3529)

SOLUTIONSincethespectraldensity

isarationalfunctionoffrequencyζ(t)satisfiestheequation

whereξ(t) isldquowhitenoiserdquowithzeroexpectationandunitvarianceThereforeconsideringatwo-dimensionalstochasticprocesswithcomponentsU1(t)=Z(t)

U2(t)= ζ(t) for the conditional probability density f(t x1 x2 τ y1 y2) of thisprocess we obtain the second Kolmogorov equation in the following form

For the stationary mode f(t x1 x2 τ y1 y2) = f(y1 y2) and the Kolmogorovequation becomes

According to the assumption of this problem it is necessary to determine themoments ml of the ordinate of Y1(t) to the fourth inclusive The requiredmoments relate to the two-dimensional probability density f(y1 y2) by

where

MultiplyingbothsidesoftheKolmogorovequationby integratingtheresultwith respect to y1 with infinite limits and taking into account that

weobtainarecursionrelationbetweenχl(y1)andχlndash1(y2)

Multiplying both sides of the last equality successively by 1 and integrating by parts and eliminating all zero terms that appear outside theintegral we get a series of equations

Setting l=1 in theseequalitieswecanexpress the fourmoments in termsof

χo(y2) Because of the normality of function Y2(τ) = ζ(τ)

Consequently all the integrals appearing in the preceding equalities can becomputedandtheresultcoincideswiththatofProblem3529whichissolvedinamorecomplicatedmanner

Example406Determinetheconditionalprobabilitydensityf(tx1hellipxnτy1hellipyn)ofamultidimensionalMarkovprocessif in thesecondKolmogorov

equationthe coefficientsbjk are constants the coefficientsai are linear functions of yj

andtherangeofyjis(ndashinfininfin)

SOLUTIONByassumptionthesolutionmustsatisfytheinitialcondition

and the condition that f vanishes as |yl| rarr infin and

foranyτWenowturnfromtheprobabilitydensityfofthesystemofrandomvariables

Y1 Y2 hellip Yn to the characteristic function

ForthispurposewemultiplybothsidesofthesecondKolmogorovequationbyexp andintegratewithrespecttoy1y2hellipynbetween infinitelimits

Since

theequationforEhastheform

LettingE=expndashVwegetforVtheequation

which according to the initial conditions for f must be solved under theconditions

Fromthegeneraltheoryitisknownthatthedistributionlawfortheprocessconsidered is normalTherefore we seek the solution for V in the form of asecond-degree polynomial of zj that is

where kjl and are real functions of τ To determine these functions wesubstitute the last expression in the differential equation forV and equate the

coefficients of equal powers of zi in the left and right sides We find

Thesystemofequationsfor isindependentofkjlandshouldbesolvedwiththe initial conditions τ = t = xj The system of equations for kjl isindependentofyjandshouldbesolvedwiththeinitialconditionsτ=tkjl=0Fromthegeneraltheoryoflineardifferentialequationsitfollowsthat andkjlarelinearcombinationsofexponentialfunctionsofformeλ(τndasht)whereλaretherootsofthecorrespondingcharacteristicequation(incaseofmultiplerootsthecoefficientsintheexponentcanbepolynomialsofτ)Thegeneralformulascanbeobtainedbymatrixoperations

Example 407 Find the conditional probability density f(t x τ y) for aprocess defined by the equation

ifαandβareconstants

SOLUTIONWeapplytheFouriermethodiefirstweseektwofunctionsψ(τ)andχ(y)whoseproductsatisfies thegivenequationindependentof theformofthe initial conditions Substituting them in the equation we get

Since the left side of the equality is independent of y and the right side isindependentofτbothsidesmustbeequaltoaconstantwhichwedenotebyλ

obtainingThefirstequationhastheobvioussolution

Thesecondequationhasasolutionvanishingatinfinityonlyfordiscretevaluesof λ = 2nβn = 0 1hellip In this case the equation for χ(y) has the solution

where are the orthogonal Laguerrepolynomialsandσ2=α22β2Since the functionsψ(τ)andχ(y) depend on theintegernthesolutionoftheinitialdifferentialequationcanbefoundasalinearcombination of the products of these functions that is

where thecoefficientscn shouldbesuch that forτ= t the function f(txτ y)becomes δ(y ndash x) that is

Todeterminetheconstantscnitissufficienttomultiplythelastequationby

and to integrate with respect to y between the limits (0 infin) Using theorthogonalityof theLaguerrepolynomialsandthepropertiesof theδ-function

wefindthatthatis

Example408Find theprobabilityW(τ) that theordinateof theprocessU(t)definedbyequationdUdt+αU=ξ(t)whereSξ(w)=c2=const =0attimeτneverexceedsthelevely=0iffort=0U(t)=ndashββgt0

SOLUTION The probability densityw(τ y)that at time τ the ordinate of thestochasticprocesswhichneverexceedsthezerolevelwilllieontheinterval(yy + dy)is defined by the second Kolmogorov equation

whichinthepresentcaseshouldbesolvedforyle0withtheconditionsw(τy)= δ(y + β) for any τ = 0w(τ 0) = 0 for any τ The required probability is

To simplify the coefficients of the equation let us introduce nondimensionalvariables

afterwhichtheequationbecomes

w(τ10)=0forτ1gt0where Solving this equation by the Fouriermethod and settingw(τ1 y1 =ψ(τ1gt)

χ(y1) we obtain for ψ(τ1) and χ(y1) the equations

Thefirstequationhastheobvioussolution andthesecondonehasfinitesolutionsatinfinityonlyifλ2=n(n=012hellip)when

where

is the Hermite polynomial Consequently the solution must be sought in theform

Since for y1 = 0 w must vanish for any τ1 the series can contain onlypolynomialsHn(y1)withoddindices(H2k+1(0)=0H2k(0)ne0foranyintegerkgt 0) Therefore the solution should be of the form

Tofindthecoefficientsa2k+1itisnecessarytofulfilltheinitialconditionthatis

Thisconditionisequivalentfortherange(ndashinfin+infin)ofy1tothecondition

MultiplyingbothsidesofthelastequalitybyH2k+1(y1)integratingwithrespectto y1 from ndashinfin to +infin and considering that

(δnn=1δnm=0fornnem)weobtain

Thus

Returningtovariablesyandτwefind

SubstitutingtheresultingseriesintheformulaforW(τ)andconsideringthat

weobtainthat

PROBLEMS

401FindthecoefficientsoftheKolmogorovequationsforann-dimensionalMarkovprocess if itscomponentsU1(t)U2(t)hellipun(t) are determinedby thesystem of equations

whereψjandφj areknowncontinuous functionsof their variables andξj(t) areindependent random functions with the properties of ldquowhite noiserdquo

402Giventhesystemofdifferentialequations

whereψjareknownfunctionsoftheirargumentsandZ(t)isanormalstationary

stochasticprocesswithspectraldensityadd to themultidimensional processU1(t)hellipUn(t) the necessary number ofcomponents so that theprocess obtained isMarkovianWrite theKolmogorovequationsforit

403SupposeU(t)astationarynormalprocessisgivenwithspectraldensity

wherecαandβareconstantsShow that U(t)can be considered as a component of a multidimensional

Markov processDetermine the number of dimensions of this process and thecoefficientsoftheKolmogorovequations

404 Determine the coefficients of the Kolmogorov equations of amultidimensional Markov process defined by the system of equations

where

andφjandψjlareknowncontinuousfunctionsoftheirarguments405TherandomfunctionsUj(t)satisfythesystemofdifferentialequations

where φj are known continuous functions of their arguments and Z(t) is astationary normal random function with rational density

wherethepolynomials

haverootsonlyintheupperhalf-planeShow that U1(t) hellip Ur(t) can be considered as components of a

multidimensionalMarkovprocessdeterminethenumberofdimensionsandthecoefficientsoftheKolmogorovequationsofthisprocess

406ShowthatiftheKolmogorovequations

whereαjαjmbjm(jm=12hellipn)areconstantsholdforamultidimensionalMarkovprocess then the stochasticprocess satisfies the systemofdifferential

equations

where

407Derivethesystemofdifferentialequationsforthecomponentsofatwo-dimensionalMarkov processU1(t)U2(t) if the conditional probability densityf(t x1 x2 τ y1 y2) satisfies the equation

408DeterminethedistributionlawfortheordinateofarandomfunctionU(t)forthestationarymodeif

whereα is aconstantφ(U) is agiven function that ensures the existenceof astationarymodeand

Solvetheproblemfortheparticularcasewhenφ(U)=β2U3409Determine thestationarydistribution lawfor theordinateofarandom

functionU(t)ifwhereφ(U)andψ(U)areknownfunctionsandξ(t)representslsquowhitenoiserdquowithzeroexpectationandunitvariance

4010 A diode detector consists of a nonlinear element with volt-amperecharacteristicF (V) connected in series with a parallel RC circuit A randominputsignalζ(t)isfedtothedetectorDeterminethestationarydistributionlawofthevoltageU(t)intheRCcircuitiftheequationofthedetectorhastheform

whereRandCareconstantsandζ(t)isanormalstationaryfunctionforwhich

Solvetheproblemfortheparticularcaseinwhich

4011DeterminethedistributionlawfortheordinateofarandomfunctionU(t) for time τ gt 0 if

4012An input signal representing a normal stochastic process ζ(t)with asmallcorrelationtimeisreceivedbyanexponentialdetectorwhosevoltageU(t)

isdefinedbytheequationwhere R C a i0 are the constants of the detector = 0 and

Usingtheapproximaterepresentation

andconsideringthat

isaδ-correlatedprocess

where

determinethestationarydistributionlawfortheordinateofU(t)4013AstochasticprocessU(t)satisfiestheequation

where φ(U) is a given function ζ(t) represents ldquowhite noiserdquo with zeroexpectation and unit variance and for a given form of the function φ(U) astationary mode is possible Determine the probability density f(y) of thestationarymode

4014ArandomfunctionU(t)satisfiestheequation

withinitialconditionsτ=tU(t)=xFindthedistributionlawfortheordinatesofthisrandomfunctionfortimeτ

getifα(t)β(t)andγ(t)areknownfunctionsoftimeandξ(t)isldquowhitenoiserdquowithzeroexpectationandunitvariance

4015The deviation of the elevator of an airplane is communicated to theautomatic pilot to eliminate the effect of wind pulsations characterized by arandomfunctionε(t)The signal is approximatelydescribedby thedifferential

equationwhereT0andi0areconstants

Determine theconditionalprobabilitydensity f(txτy) of theordinate oftherandomfunctionΔ(t)iftheexpectation =0andonemayapproximatelyconsiderthat andΔ=xforτ=t

4016Theincomingrandomperturbationattheinputofasystemofsecondorder is described by ζ(t)

Determine theconditionaldistribution lawof theordinateof the stochastic

processU(t)attimeτgetifattimetU(t)=x Kζ(τ)=c2δ(τ))chkareknownconstants

4017 The equation defining the operation of an element of a system of

automaticcontrolhastheformwhereαandcareconstantsand

Write the Kolmogorov equation for the determination of the conditionalprobabilitydensityf(txτy)

4018 A moving charged particle is under the influence of three forcesdirected parallel to the velocity vectorU(t) the forces created by the electricfieldofintensityξ(t)theacceleratingforcecreatedbythefieldwhoseintensitycanbetakeninverselyproportionaltothevelocityoftheparticleandthefrictionforces proportional to the velocity The motion equation has the form

Findtheprobabilitydensityf(txτy)forthemagnitudeofthevelocityU(t)ifα

βandγareconstantsand themassoftheparticleism

4019 A radio receiver can detect a random input noiseU(t) only if theabsolutevalueofthesignalisgreaterthanthesensitivitylevelofthereceiveru0Determine the probability W(T) that during time T no false signal will bereceived ifU(t) is a normal stochastic processwith zero expectation andwithcorrelationfunctionwhereu0αandσareconstantsandU(t)=0fort=0

4020AradioreceivercandetectarandominputnoiseU(t)ifthesignal(notits absolute value) is greater than the sensitivity level u0 of the receiverDetermine the probability W(T) that during time T no false signal will bereceived ifU(t) is a normal stochastic processwith zero expectation andwithcorrelationfunctionwhereu0αandσareconstantsandU(t)=0fort=0

1 Weinberger H F First Course in Partial Differential Equations Waitham MAss Blaisdell

Publishing Company 1965 and Petroviskii IG Partial Differential Equations Philadelphia WBSaundersCompany1967

2ThesecondKolmogorovequationissometimescalledtheFokker-PlanckequationorFokker-Planck-KolmogorovequationsincebeforeitwasrigorouslyprovedbyKolmogorovithadappearedintheworksofthesephysicists


41 DETERMINATIONOFTHEMOMENTSOFRANDOMVARIABLESFROMEXPERIMENTALDATA

BasicFormulas

The approximate values of the moments of random variables obtained byprocessing the experimental data are called estimates (fitting values) of thesevariables and are denoted by the same symbols as the estimated numericalcharacteristics of random variables but with a tilde above (for example

andsoforth)Thesetofvalues(x1x2hellipxn)forarandomvariableXobtainedinnexperimentsiscalledasampleofsizen It is assumed that the experiments are performed independently under thesame conditions If the sample size n tends to infinity the estimate shouldconvergeinprobabilitytotheparameterbeingestimatedTheestimateiscalledunbiased if for any sample size its expectation coincides with the requiredparameter The unbiased estimate for the expectation is the arithmetic mean

whereC is an arbitrary number introduced for convenience in computations(ldquofalsezerordquo)

Iftheexpectedvalueisunknowntheunbiasedestimateofthevariancewillbe

Iftherandomvariableconsideredisnormallydistributedthentheunbiasedestimateofthestandarddeviationisgivenbytheformula

where

TABLE23

ThevaluesofthecoefficientknareincludedinTable23Iftheexpectationisknowntheunbiasedestimateofthevarianceis

Ifx1y1hellipxnynarethevaluesoftherandomvariablesXandYobtainedas a result of n independent experiments that are performed under identicalconditionstheunbiasedestimateofthecovarianceoftheserandomvariablesis

forunknownexpectationsXandY

forknownexpectationsTheestimateofthecorrelationcoefficientcanbefoundfromtheformula

Foralargesamplesizetheelementsofthestatisticalseriesarecombinedingroups(classes)byrepresentingtheexperimentaldataintheformofanorderedarray(Table24)

TABLE24

In this case the estimates for the expectation variance and moments ofhigherorderareapproximatelydeterminedbytheformulas

ormoreprecisely(takingintoaccounttheSheppardcorrections)by

wherehistheclassintervallength


Example411TodeterminetheprecisionofameasuringinstrumentwhosesystematicerrorispracticallyzerooneperformsfiveindependentmeasurementswhoseresultsaregiveninTable25

Determinetheunbiasedestimateforthevarianceoferrorsifthevalueofthequantitybeingmeasuredis(a)knowntobe2800m(b)unknown

TABLE25

SOLUTIONThevalueof thequantitybeingmeasured is Therefore in (a)

the unbiased estimate of the variance is determined by the formula

Ifthevalueofthemeasuredquantityisunknownitsestimateis

Thusin(b)theunbiasedestimateis

InasimilarwayonecansolveProblems411to4114and4113to4116

Example412 To determine the estimates of the standard deviation of theerrorsgivenbyameasuring instrumentwhosesystematicerrorsarepracticallyzerooneperformsfive independentexperimentswhoseresultsare included inTable26

TABLE26

Toprocessthedataobtainedinmeasurementsthefollowingformulasfortheunbiasedestimatesareused

Find 1and 2 anddetermine thevarianceof theseestimates if theerrors

obeyanormaldistributionlaw

SOLUTIONFillinginTable27andsummingbycolumnsweobtain

TABLE27

Theobtainedestimates 1and 2arerandomvariableswhoseexpectationisM[ 1] = M[ 2] = σ To find the variance 1 we have

Forthevarianceoftherandomvariable 2wehave

where Let Since zi is a linear function of normal

random variables it also obeys a normal distribution law with parameters

Therefore

where(jnei)

Passingtopolarcoordinateswefind

Here

Finallyweget

Theratiobetweenthevariancesfortherandomvariables 1and 2fordifferentnareshowninTable28

TABLE28

The solution for this example implies that the estimate of σ given by theformula

hasasmallervariancethantheresultobtainedfromtheformula

thatistheestimate 1ismoreefficientSimilarlyonecansolveProblems4174112and4120

Example413Fromthecurrentproductionofanautomaticboringmachineasampleof200cylindersisselectedThemeasureddeviationsofthediametersofthesecylindersfromtheratedvaluearegiveninTable29

Determine the estimates for the expectation variance asymmetry and theexcessofthesedeviations

SOLUTIONTosimplifytheinterrcdiarycalculationsweintroducetherandomvariable

where as ldquofalse zerordquowe takeC = 25microns and the classwidth ish= 5microns

TABLE29

Let us determine the estimates of the first four moments of the randomvariable by considering the Sheppard corrections The calculations aresummarizedinTable30

TABLE30

TakingintoaccounttheSheppardcorrectionsweobtain

For the same variables but without considering the Sheppard corrections wehave(seeExamples432and434)


PROBLEMS

411In12independentmeasurementsofabaseoflength23238mwhichwereperformedwiththesameinstrument thefollowingresultswereobtained2325023248232152325323245232302324823205232452326023247and23230mAssumingthat theerrorsobeyanormaldistributionanddo not contain systematic errors determine the unbiased estimate for thestandarddeviations

412 The following are the results of eight independent measurementsperformedwithaninstrumentwithnosystematicerror369378315420385401 372 and 383mDetermine the unbiased estimate for the variance of theerrors inmeasurements if (a) the length of the base that is beingmeasured isknown =375m(b)thelengthofthemeasuredbaseisunknown

413 In processing the data obtained in 15 tests performed with a modelairplane the following values for its maximal velocity were obtained 42224187 4256 4203 4258 4231 4315 4282 4383 4340 4113 41724135 4413 and 4230 msec Determine the unbiased estimates for theexpectationandstandarddeviationof themaximalvelocityassumed toobeyanormaldistributionlaw

414 In processing the data of six tests performed with a motorboat thefollowingvaluesforitsmaximalvelocitywereobtained2738303735and31msec Determine the unbiased estimates for the expectation and standarddeviation of themaximal velocity assuming that themaximal velocity of theboatobeysanormaldistributionlaw

415The sensitivity of a television set to video signals is characterizedbydatainTable31

TABLE31

Find the estimates for the expectation and standard deviation of thesensitivityoftheset

416Anumbernofindependentexperimentsareperformedtodeterminethefrequency of an event A Determine the value of P(A) that maximizes thevarianceofthefrequency

417 A number n of independent measurements of the same unknownconstantquantityareperformedTheerrorsobeyanormaldistributionlawwithzeroexpectation

Todetermine the estimatesof thevariancebyusing the experimentaldatathefollowingformulasareapplied

Findthevarianceoftherandomvariables and 418TheexperimentalvaluesofarandomvariableXaredividedintogroups

Theaveragevalue forthejthgroupandthenumberofelementsmjinthejthgroupareinTable32

TABLE32

Findtheestimatesfortheasymmetrycoefficientandtheexcess419 A sample x1 x2hellip xn selected from a population is processed by

differences in order to determine the estimates for the variance The formulaused for processing the results of the experiment is

How large should k be so that is an unbiased estimate of if therandomvariableXisnormal

4110Letx1x2hellipxnbetheoutcomesofindependentmeasurementsofanunknown constant The errors in measurements obey the same normaldistribution law The standard deviation is determined by the formula

where

Determinethevalueofkforwhich isanunbiasedestimateofσ4111Independentmeasurementsofaknownconstantxarex1x2hellip xn

TheerrorsobeythesamenormaldistributionlawForprocessingtheresultsoftheseobservationsinordertoobtaintheestimatesforthestandarddeviationof

errorsthefollowingformulaisusedHowlargeshouldkbesothattheestimatesareunbiasedfor(a)thestandard

deviationoftheerrors(b)thevarianceoftheerrors4112Independentmeasurementsx1x2hellipxnwithdifferentaccuraciesof

the same unknown constant are made The estimate of the quantity being

measuredisdeterminedfromtheformulaHowlargeshouldAjbesothatthevarianceof isminimalif thestandard

deviationoftheerrorsofthejthmeasurementisσj4113A systemof two randomvariableswith a normal distribution in the

planeissubjectedtonindependentexperimentsinwhichthevalues(xkyk)(k=12hellipn)ofthesevariablesaredeterminedTheprincipaldispersionaxesareparallel to the coordinate axes Determine the unbiased estimates for theexpectationandthestandarddeviationsofthesevariables

4114SolveProblem4113fortheresultsoftheindependenttrialsgiveninTable33

4115 Under the conditions of Problem 4113 find the estimates for theparametersoftheunitdispersionellipseifbeforetheexperimentsthedirectionoftheprincipalaxesisunknown

4116SolveProblem4115for theresultsof16 independent trialsgiven inTable34

TABLE33

TABLE34

4117Asamplex1x2hellipxnselectedfromanormalpopulationisprocessedto determine the estimates for the standard deviation by the formula

where

How large should k be so that is an unbiased estimate of the standarddeviationσ

4118Fromatableofrandomnumbers150two-digitnumbers(00istakenfor100)areselectedThesenumbersaredividedintointervalsof10(Table35)

TABLE35

Construct the histogram and the graph of the frequency count Find theestimatesfortheexpectationandvariance

4119Withtheaidofatableofrandomone-digitnumbers250sumsoffivenumberseachareformedThenumbersaredistributedintoclassesasindicatedinTable36(ifthenumbercoincideswiththelimitofaclassfrac12isaddedtothetwo adjacent classes) Construct the histogram and find the estimates for theexpectationandvariance

TABLE36

4120 To determine the value of an unknown constant n independentmeasurements are performed The systematic errors inmeasurements are zeroandtherandomerrorsarenormallydistributedThefollowingtwoformulasareused to find the estimated variances

Are and unbiased estimates of the variance Which of these twoformulasgivesamoreaccuratevalueforthevariance

42 CONFIDENCELEVELSANDCONFIDENCEINTERVALS

BasicFormulas

A confidence interval is an interval that with a given confidence level αcoversaparameterΘtobeestimated

The width of a symmetrical confidence interval 2ε is determined by thecondition

where is the estimate of parameter Θ and the probability

isdeterminedbythedistributionlawfor Ifx1x2hellipxn is a sample fromanormalpopulation then the confidence

levelisdeterminedbytheformulas(a)fortheexpectationinthecasewhenσisknown

forunknownσ

where

isStudentrsquosdistributionlawand

Thevaluesof tα aregiven inTable16T2whose entries are thenumberofdegreesoffreedomk=nndash1andtheconfidencelevelα

(b)Forthestandarddeviation

where

Thevaluesoftheintegral aregiveninTable20TThe confidence interval for where the probabilities of its

lyingentirelytotherightandentirelytotheleftofthetruevalueareboth(1ndashα)2 is determined by the formula

Inordertofindγ1andγ2foragivenconfidencelevelσandk=nndash1degreesoffreedomonemayuseTable19Tor18T

For anexponentialdistribution law the confidence interval for expectation(v1 v2 ) is given by the expression

Fromthis

The values for and are determined from Table 18T for theprobabilitiesδand1ndashδrespectivelyandfork=2ndegreesoffreedom

For a sufficiently large sample size (n gt 15) the limits of the confidenceinterval for are calculated approximately by the formulas

whereε0isthesolutionoftheequationα=Φ(ε0)IffromthesamepopulationthereareselectedNsampleseachofsizenif

theeventwhoseprobabilityofrealizationobeysaPoissondistributionoccursmjtimes(j=12hellipN)inthejthsampleandtheexpectedvalueoftheparameterisgivenby the formula thenforatildegt0 the limitsof theconfidence interval are determined from the relation

thatistheupperandlowerlimitsareequalto

respectivelywhere and given δ are chosen from Table 18Tbeing taken for degrees of freedom and for

degreesoffreedomForatilde=0thelowerlimitbecomeszeroandtheupperlimitis 2NwhereisfoundfromTable18Tfork=2andlevel For a sufficiently large k (practically greater than 30) the limits of the

confidenceintervalaredeterminedapproximatelybytheformulas

whereε0isthesolutionoftheequationα=Φ(ε0)Ifinnindependenttrialsacertaineventoccursexactlymtimes(0ltmltn)

thelimitsp1p2oftheconfidenceintervaliftheprobabilityofoccurrenceofthisevent is p are determined from the equations

Theseequationscanbesolvedapproximatelywiththeaidoftheincompleteβ-functionInTable30Tthevaluesofp1andp2aregivenfordifferentmandnandtwovaluesofthelevelα095and099

Fornsufficientlylargeonecanwriteapproximately

where mnandεisthesolutionoftheequation

Abetterapproximationisgivenbytheformulas

and

oneofwhichunderestimates the intervalwhile theotheroverestimates itbyaquantityofthesameorderofmagnitudeε0isthesolutionofα=Φ(ε0)

Ifm=0thenp1=0and

Ifm=nthenp2=1and

The confidence interval for the correlation coefficient whose estimate isobtained from a normal sample of size n can be expressed approximately interms of auxiliary random variable whoselimits(ZHZB)oftheconfidenceintervalaredeterminedbytheformulas

where ε0 is the solution of the equation

(the

valueofthisquantityisdeterminedfromTable31T)and BythevaluesZHandZBfoundfromTable31Tortheformular= tanhz

onecanfindthelimitsoftheconfidenceintervalforrInthecaseoflargen(ngt50)andsmall ( lt05) thelimitsrHrBof theconfidenceintervalforrare

givenapproximatelybywhereε0isthesolutionofthequationα=Φ(ε0)


Example421Theaveragedistancemeasuredfromareferencepointinfourindependenttrialsis2250mThemeanerrorofthemeasuringinstrumentisE=40mGiven theconfidence level95percent find theconfidence interval forthequantitymeasured

SOLUTION The probability of covering the true value of the measuredquantitybytheinterval( ndashε +ε)withrandomendpointsforaknownE isdetermined by the formula

where is the standard deviation of the random variable

Solving the equation bymeans of Table 11T wefindthat

Fromthisthelimitsoftheconfidenceintervalwillbe


Example422Thestandarddeviationofanaltimeterisσ=15mHowmanyaltimetersshouldtherebeonanairplanesothatwithconfidencelevel099themeanerror inaltitude isnotgreater thanndash30m if theerrorsgivenby thealtimetersarenormallydistributedandtherearenosystematicerrors

SOLUTIONTheassumptionsoftheproblemcanbewrittenas

Therandomvariable

isalinearfunctionofnormallydistributedrandomvariablesandhenceitalsoobeysanormaldistributionwithparameters

Then

Solvingtheequation

wefindfromTable8Tthat

ThusthenumberofaltimetersontheairplaneshouldbeatleasttwoProblems427and4211canbesolvedsimilarly

Example423 Incontrol testsperformedwith16 lightbulbs estimates fortheirexpected lifetimeand thestandarddeviationwere found =3000hoursand =20Ifthelifetimeofeachbulbisanormalrandomvariabledetermine(a) theconfidenceintervalfor theexpectationandthestandarddeviationif theconfidence level is 09 (b) the probabilitywithwhich one can assert that theabsolutevalueoftheerror willbeatmost10hoursandtheerrorwillbelessthantwohours

SOLUTION (a) To determine the limits of the confidence interval for theexpectationwemakeuseoftheequation

In Table 16T for k = n ndash 1 and α = 09 we find that

hence it follows that

hours

Therefore the upper and lower limits of the confidence interval for are3000 + 8765 = 3008765 hours and 3000 ndash 8765 = 2991235 hoursrespectively

To determine the limits of the confidence interval for σ we make use ofTable19TTheentriestothistablearek=nndash1andtheconfidencelevelisαFork=15andα=09wehave

Consequentlyforaconfidencelevel09thevaluesofαcompatiblewiththeexperimentaldataliewithinthelimits0775 =1550hoursto1437 =2874hours

(b) The probability for the inequality ndash 10 hours lt ndash lt 10 hours isdetermined by Studentrsquos distribution

FromTable16Tfor andk=nndash1=15wefindthatα=093

The chi-square distribution permits us to determine the probability for theexistence of inequality ndash 2 hours lt ndash σ lt 2 hours

Forq=ε =220=01andk=nndash1=15degreesoffreedomwefindfromTable20Tthatα=041

Following this solution one can solve Problems 422 to 425 and 428 to4210

Example424 A random variableT obeys an exponential distribution lawwithaprobabilitydensityf(t)=1 expndasht

Theestimatefortheparameter isdeterminedbytheformula

Express in terms of the limits of the confidence interval for so that if the confidence level α =

09andnequals3510203040

SOLUTIONBytheassumptionsmadeinthisexample

Rewritingtheinequalitiesinthisexpressionleadsto

TherandomvariableU=2n hasachi-squaredistributionwith2ndegreesof freedom and for n gt 15 the random variable has anapproximately normal distribution with and σz = 1Therefore in the first case (for n lt 15) we have

Afterdetermining and fromTable18T(for2ndegreesoffreedomandprobabilitiesδand1ndashδ)wecalculatev1andv2(seeTable37)

TABLE37

Inthesecondcase(ngt15)accordingtotheformulasatthebeginningofthissolutionwehave(seeTable38)

TABLE38

Thequantityε0isdeterminedfromTable8Tforthelevelα=09InFigure35thereisgiventhegraphrepresentingυ1andυ2asfunctionsofn

fortheconfidencelevelα=09

Example 425 Three types of devices (A B and C) are subjected to 50independent trials during a certain time interval the numbers of failures arerecordedasinTable39FindthelimitsoftheconfidenceintervalsfortheTABLE39

expectednumberof failuresofeach typeduringa selected time interval if theconfidence level α = 09 and the number of failures for each type obeys a

Poissondistributionlawduringthisinterval

FIGURE35

SOLUTIONTodeterminethelimitsoftheconfidenceintervalforthedevicesoftypeAwemakeuseofachi-squaredistributionFromTable18Tfork=24degreesoffreedomandprobability(1+α)2=095wefind =138fork=26andprobabilityδ=(1ndashα)2=005wefind =389

The upper limita2 and the lower limita1 of the confidence interval forādevicesoftypeAareequalto

To determine the limits of the confidence interval for the expected number ofdevicesoftypeBthatfailedonealsoshouldusethechi-squaredistributionfork=180andk=182degreesoffreedomTable18Tcontainsthedataonlyfork=30Thereforeconsideringthatforanumberofdegreesoffreedomgreaterthan30achi-squaredistributionpracticallycoincideswithanormalonewehave

For devices of typeC and therefore the lower limit of theconfidenceintervaliscertainlyzeroFromTable18Tfork=2andprobability1ndashα=01wedetermine =46andcalculatethevaluefortheupperlimita2= 2N=46100=0046

Example 426 Ten items out of thirty tested are defective Determine thelimitsoftheconfidenceintervalfortheprobabilityofadefectiftheconfidencelevel is 095 and the number of defective itemsobeys a binomial distributionComparetheresultsoftheexactandapproximatesolutions

SOLUTIONTheexactsolutioncanbeobtaineddirectlyfromTable30TForx=10nndashx=20andaconfidencelevelequalto95percentwehavep1=0173p2=0528

Forlargenp(1ndashp)theequationsfromwhichwedeterminethelimitsoftheconfidence interval for p can be written approximately by using the normaldistribution

Fromthis

where =mn=13andthequantityεe0canbedeterminedfromTable8Tforlevel α = 095

Anapproximationofthesamekindgivestheformula

whichwhenappliedleadsto

Byarougherapproximationp1andp2canbefoundifoneconsidersthatthefrequency isapproximatelynormallydistributedaboutpwithvariance (1ndash) Inthiscase

whereεisthesolutionoftheequation UsingTable8Tforα=095weget

henceitfollowsthatp1asymp0333ndash0169=0164p2asymp0333+0169=0502

Example427To study themechanicalpropertiesof steel 30 independentexperiments areperformedbasedon theiroutcomes estimates aredeterminedfor the correlation coefficients 12 = 088 and 13 = 040 characterizing therelationoftheenduranceleveltotheresistanceandfluiditylevelsrespectivelyDeterminethelimitsoftheconfidenceintervalforr12andr13iftheconfidencelevelis095

SOLUTION For a large sample size n and small values of the correlationcoefficient r its estimate has a distribution approximately normal with

expectation andstandarddeviation

Takingrasymp wehaveFromTable8Tfortheconfidencelevelα=095wefindε0=196(εobeing

thesolutionoftheequationα=Φ(ε0))andtheconfidenceinterval(084092)forr12

(018062)forr13Theconfidenceintervalobtainedcanbecorrectedifwetransform sothatσ

is independent of r This leads to a new random variable

whosedistributionisapproximatelynormalevenforsmalln

Inthiscase

and

Using Table 31T we determine the confidence interval for the randomvariableZ

088(10141768)for 12040(00530808)for 13

UsingTable31Twefindtheconfidenceinterval

(077094)for 12(005067)for 13

PROBLEMS

421 A constant quantity is measured 25 times with an instrument whosesystematicerroriszeroandrandomerrorsarenormallydistributedwithstandarddeviation E = 10 m Determine the limits of the confidence interval for thevaluesofthequantitybeingmeasurediftheconfidencelevelis099and =100m

422 The results of measurements not containing systematic errors arewritten in the formof a statistical series (Table40)The errorsobeyanormaldistribution Determine the estimate of the quantity being measured and thelimitsoftheconfidenceintervalforaconfidencelevel095

TABLE40

423 From the results of 40 measurements of a base of constant lengthestimatesofthelengthandthestandarddeviationarefound =10400mand

x=85mTheerrorsobeyanormaldistributionlawFindtheprobabilitiesthattheconfidenceintervalswithrandomlimits(0999 1001 )and(095 105 )willcovertheparameters andσxrespectively

424Theresultsof11measurementsofaconstantquantityare included inTable 41 The errors are normally distributed and the systematic errors areabsent

TABLE41

Determine (a) the estimatesof the lengthbeingmeasuredand the standarddeviation (b) theprobability that theabsolutevalueof theerror in finding theexactvalueislessthan2percentof (c)theprobabilitythattheabsolutevalueoftheerrorinthestandarddeviationislessthan1percentof

425Asaresultof100experimentsithasbeenestablishedthattheaveragetimenecessaryfortheproductionofoneitemis =05andthat ωAssumingthatthetimetoproduceanitemisanormalrandomvariablefindthelimitsofthetruevaluesof andσω forconfidence levels85percentand90percentrespectively

426Theestimateforthevelocityofanairplaneobtainedfromtheresultsoffivetrialsis =8703msecFind the95percent-confidence interval if it isknown that the dispersion of the velocity obeys a normal distribution withstandarddeviationEv=21msec

427The depth of a sea ismeasuredwith an instrumentwhose systematicerror is zero and the random errors are normally distributed with standarddeviationE=20mHowmanyindependentmeasurementsshouldbeperformedtofindthedepthwithanerrorofatmost15miftheconfidencelevelis90percent

428Findforconfidencelevel09theconfidencelimitsforthedistancetoareference point and the standard deviation E if in 10 independentmeasurementstheresultsgiveninTable42wereobtainedandtheerrorsobeyanormaldistribution

TABLE42

429Assume that five independentmeasurementswith equal accuracy areperformed to determine the charge of an electron The experiments give thefollowing results (in absolute electrostatic units)

Findtheestimateforthemagnitudeofthechargeandtheconfidencelimitsofaconfidencelevelof99percent

4210 From the results of 15 independent equally accurate measurementsthere were derived the following values for the estimate of the expectedmagnitudeandthestandarddeviationofthemaximalvelocityofanairplane=4247msecand v=87msec

Determine (a) the confidence limits for the expectation and the standarddeviationiftheconfidencelevelis09(b)theprobabilitieswithwhichonemayassertthattheabsolutevalueoftheerrorin andσvdoesnotexceed2msec

4211 The arithmetic mean of the results of independent measurementsperformed with n range finders is taken as an estimate of the distance to anavigational marker The measurements contain no systematic errors and therandomerrorsarenormallydistributedwithstandarddeviationE=10mHowmany range finders should there be so that the absolute value of the error indeterminingthedistancewillbeatmost15mwithprobability09

4212ItisknownthatameasuringinstrumenthasnosystematicerrorsandtherandomerrorsofeachmeasurementobeythesamenormaldistributionlawHow many measurements should be performed to find the estimates for thestandarddeviationsothatwithconfidencelevelof70percenttheabsolutevalueoftheerrorisatmost20percentof

4213 The systematic errors of an instrument are practically zero and therandomerrorsarenormallydistributedwithstandarddeviationE=20m It isnecessarythatthedifferencebetweentheestimateofthemeasuredquantityanditstruevalueisatmost10mWhatistheprobabilitywithwhichthisconditionwill be satisfied if the number of observations is 3 5 10 25 (construct the

graph)4214Theestimateforameasuredquantityisgivenbytheformula

Theresultsof individualmeasurementsobeythesamenormaldistributionlawFind the limits of the confidence interval for level 09 with the followingconditions(a)σ=20mn=351025(b) =20mn=351025

4215TenidenticaldevicesaretestedTheinstantswheneachofthemfailedarerecordedTheresultsoftheobservationsareinTable43

DeterminetheestimatefortheexpectedtimetfornonstopTABLE43

operationofadeviceandtheconfidenceintervalfortiftheconfidencelevelis09andtherandomvariableTobeysanexponentialdistribution

4216ArandomlyselectedsampleofeightdevicesissubjectedtoreliabilitytestsThenumbers of hours duringwhich the devices operatewithout failuresare 100 170 400 250 520 680 1500 and 1200 Determine the 80 per centconfidenceintervalfortheaveragedurationofoperationifthereliableoperationtimeobeysanexponentialdistributionlaw

4217Theprobabilitydensity for the timebetweensuccessive failuresofa

radio-electronicdeviceisgivenbytheformulawheretistheoperatingtimebetweentwosuccessivefailures istheexpectedvalue of the random variableT which is the expected time duringwhich thedevice operates in good condition (called in reliability theory ldquothe expectedlifetimerdquo)

For thedeterminationof theestimatesof theparameter 25 failureswereobservedandthetotaldurationofthereliabletimefromthestartoftheteststothelastfailureturnedouttobe =1600hours

Find the limitsof theconfidence interval for theparameter according to

theresultsoftheseexperimentsiftheconfidencelevelisα=084218Todeterminethetoxicdoseacertainpoisonisadministeredto30rats

eight of which die Determine the limits of the confidence interval for theprobabilitythatthegivendosewillbefataliftheconfidencelevelis095andifthenumberof fatal outcomes in this experimentobeys abinomial distributionlaw

4219 In 100 independent trials a certain event A is observed 40 timesDeterminethelimitsoftheconfidenceintervalfortheprobabilityofoccurrenceof this event if the confidence levels are 095 and 099 and the number ofoccurrencesofAhasabinomialdistribution

4220 Ten devices are tested and no failures are observed Determine thelimitsoftheconfidenceintervalinthecasesinwhichtheconfidencelevelsare0809and099ifthefailureshaveabinomialdistribution

4221AmarksmanAscoresfivehitsin10shotsandBscores50hitsin100shotsbothmarksmenfireshotsatthesametargetDeterminethelimitsoftheconfidenceintervalfortheprobabilitiesthateachmarksmanscoresahitinoneshotiftheconfidencelevelis099andthehitsobeyabinomialdistribution

4222SixidenticaldevicesaretestedDuring15hoursoftests12failuresarerecordedFindthelimitsof theconfidenceintervalfor theexpectednumberoffailuresduring15hoursoperationofsuchadeviceiftheconfidencelevelis09andthetesteddevicesobeyaPoissondistribution

4223 The number of particles recorded by a counter in the Rutherford-Chadwick-Ellisexperimentduringeveryoneof2608intervalsof75seceachisgiven in Table 44 Assuming that the number of particles obeys a Poissondistributiondeterminethelimitsoftheconfidenceintervalfortheparameterofthisdistributioncorresponding toan intervalof75secand to theconfidencelevel09999

TABLE44

4224 In analyzing the amount of dodder in seeds of clover it has beenestablished that a sample of 100 g contains no dodder seedsFind the 99 percent-confidence interval for the average amount of dodder in a sample thatweighs100giftheamountofdodderobeysaPoissondistribution

4225 From the results of 190 experiments performedwith itemsmade ofTypeAsteelestimateswerefoundforthecorrelationcoefficients 12=05513=030 14=037characterizingthedependenceofthecoerciveforceonthegrainandcarbonandsulphurcompositionsrespectivelyDeterminethelimitsoftheconfidence intervals for thecorrelationcoefficients if theconfidence levelsare099and095andiftherandomvariableshaveanormaldistribution

4226Inacertainexperiment25pairsofvalueswereobtainedforasystemofrandomvariables(XY)withanormaldistributionWith theseexperimentaldatatheparametersofthissystemwereestimated =105 =74 x=20y = 100 xy = 062 Find the limits of the confidence intervals for theparametersofthesystem(XY)iftheconfidencelevelis09

43 TESTSOFGOODNESS-OF-FIT

BasicFormulas

Thetestsofgoodness-of-fitpermitestimationoftheprobabilitythatacertainsample does not contradict the assumption made regarding the form of thedistribution lawofaconsidered randomvariableFor thispurposeoneselectssome quantity κ representing the discrepancy measure of the statistical andtheoreticaldistributionlawsanddeterminesforitavalueκαsuchthatP(κltκα)=αwhereα isasufficientlysmallquantity(significancelevel)whosevalueis

determined by the nature of the problem If the experimental value of thediscrepancy measure κq is greater than κα the deviation from the theoreticaldistributionlawisconsideredsignificantandtheassumptionregardingtheformof the distribution is disproved (the probability of disproving a correctassumptionwithregardtotheformofthedistributioninthiscaseisequaltoα)Ifκqleκαthentheexperimentaldataagreewiththehypothesismadeabouttheformofthedistributionlaw

Thetestofthehypothesisaboutthecharacterofthedistributionbymeansofgoodness-of-fitprocedurescanbeperformedinanotherorderaccordingtothevalueκqonedeterminestheprobabilityαq=P(κltκq)Ifαqltαthedeviationsare significant if αq geα the deviations are insignificant The values αq verycloseto1(verygoodfit)correspondtoaneventwithverysmallprobabilityofoccurrenceandindicatethatthesampleisdefective(forexampleelementswithlargedeviationsfromtheaverageareeliminatedfromtheinitialsamplewithoutfurtherreason)

Indifferenttestsofgoodness-of-fitdifferentquantitiesaretakenasmeasuresofdiscrepancybetweenthestatisticalandtheoreticaldistributions

In the chi-square tests (the Pearson tests) the discrepancy measure is thequantity χ2 whose experimental value is given by the formula

where l is the number of classes into which all experimental values ofX aredivided n is the sample sizemi is the number in the ith class and pi is theprobability computed from the theoretical distribution law that the randomvariableXisintheithclassinterval

Fornrarrinfinthedistributionof regardlessofthedistributionoftherandomvariable X tends to a chi-square distribution with k = l ndash r ndash 1 degrees offreedomwhereristhenumberofparameterscomputedaccordingtothegivensampleofthetheoreticaldistributionlaw

The values of the probabilities as functions of and k aregiveninTable17T

Toapplythechi-squaretestinthegeneralcaseitisnecessarythatthesamplesizen and class numbersmi be sufficiently large (practically it is consideredsufficientthatn~50ndash60mi~5ndash8)

TheKolmogorovtestofgoodness-of-fitisapplicableonlyiftheparametersofthetheoreticaldistributionlawarenotdeterminedbythedataofthesample

The biggest value D of the absolute value of the difference between thestatistical and theoretical distribution functions is selected as the discrepancymeasure of the statistical and theoretical distribution laws The experimental

valueDqofDisdeterminedbytheformulawhere and F are the statistical and the theoretical distribution functionsrespectively

Asnrarrinfinthedistributionlawforλ= regardlessoftheformofthedistributionoftherandomvariableXtendstotheKolmogorovdistributionThevaluesoftheprobabilitiesαq=P(DgeDq)=P(λ)=1ndashK(λ)areincludedinTable25T

The Kolmogorov test is also a statistical test of the hypothesis that twosamplesofsizen1andn2arisefromasinglepopulationInthiscaseαq=P(λ)where P(λ) is given in Table 25T but

where 1(x)and 2(x)arethestatisticaldistributionfunctionsforthefirstandsecondsamples

TheformofthetheoreticaldistributionischoseneitheronthebasisofdataabouttherandomvariablesselectedorbyqualitativeanalysisoftheformofthedistributionhistogramIftheformofthedistributioncannotbeestablishedfromgeneralconsiderationsthenitisapproximatedbyadistributionwhosefirstfewmoments are the same as the estimates obtained from the sample Forapproximating expressions one can use Pearsonrsquos curves (Gnedenko andKhinchin 1962) which consider the four first moments or the infinite Edge-worthseries(GnedenkoandKhinchin1962)Hereforasmalldeviationofthestatistical distribution from the normal one can retain only the first termsforming a Charlier-A series

where φ2(Z) φ 3(Z) are the second and third derivatives of the normal

probability density φ(z) z = (x ndashM[x]) is the estimate for

asymmetry istheestimateforexcessand and4 are the estimates for the second third and fourth central moments

respectively

ThevaluesofФ(z)φ2(z)φ3(z)aregiveninTables8Tand10TThechi-squaretestalsopermitsustotesthypothesesabouttheindependence

oftworandomvariablesXandYInthiscase isdeterminedbytheformula

wherehijisthenumberofcasesinwhichthevaluesX=xiY=yjareobserved

simultaneouslyhi0being thenumberofcases inwhich thevalueX=xi isobservedh0j is thetotalnumberofcasesinwhichthevalueY=yj isobservedand landmare thenumbersofvaluesassumedbythevariablesXandY

The number k of degrees of freedom necessary for the calculation of theprobability isgivenbytheformula


Example431A radioactive substance isobservedduring2608equal timeintervals (each 75 sec) For each interval the number of particles reaching acounter is recorded The numbersmiof time intervals during which exactly iparticlesreachedthecounteraregiveninTable45TABLE45

Test using the chi-square test the hypothesis that the data agreewith thePoissondistributionlaw

Thesignificancelevelshouldbetakenas5percent

SOLUTIONUsingthedatawecomputetheestimateatildeof theparameteraofthePoissondistributionbytheformula

where For the functionP(i atilde) =pi we compute using Table 6T the theoretical

probabilitiespithatiparticleswithPoissondistributionreachthecounterAsaresultofinterpolationbetweena=3anda=4weobtainthevaluespiandnpiwhicharegiveninTable46

We compute the values of by performing the calculations in Table 46

Since the number of degrees of freedom is k = l ndash r ndash 1 where the totalnumberofintervalsisl=11andthenumberofparametersdeterminedfromthedataisr=1(theparametera)wehaveFromTable17Twefindfork=9and =1305 theprobability

that the quantity χ2 will exceed We obtain

TABLE46

Since αq gt α = 005 the deviations from the Poisson distribution areinsignificant


Example432Asampleof200itemsisselectedfromthecurrentoutputofaprecision automatic lathe The dimension of each item is measured with aprecisionof1micronThedeviationsxi (fromthenominaldimension)dividedintoclassesandthenumbersintheclassesandtheirfrequenciesparegiveninTable47

TABLE47

Estimatewith the aid of the chi-square test the hypothesis that the sampledistribution obeys a normal distribution law for a significance levelα = 5 percent

SOLUTIONWedeterminethevalues ofthemidpointsoftheintervalsand

find the estimates for the expectation and variance by the formulas

ThecomputationsaresummarizedinTable48

TABLE48

Thetheoreticalprobabilitiespithatthedeviationslieontheintervals(xixi+

1)arecomputedbytheformulawhereziistheleftlimitoftheithintervalmeasuredwithrespectto in units

Herethesmallestzi=z0=206isreplacedbyndashinfinandthelargestz11=309by+infin

ThevalueoftheLaplacefunctionФ(z)isfoundfromTable8TTheintervali=10becauseofitssmallnumberisattachedtotheintervali=9TheresultsofthecomputationsaregiveninTable48

Wefindthat

Thenumberofdegreesoffreedomis

since because of the small numbers in the last two classes the 9th and 10thclassesareunited

FromTable17Tforthevalues andkwefindαq=P(χ2ge )=0313Thehypothesisonthenormalityof thedeviationsfromthenominaldimensiondoesnotcontradicttheobservations

Problems 436 437 439 4311 4313 to 4321 4324 and 4325 can besolvedinasimilarmanner

Example 433 The results xi of several measurements (rounded-off to 05mm)of1000itemsaregiveninTable49

TABLE49

(miisthenumberofmeasurementsgivingtheresultxi)ByusingtheKolmogorovtestofgoodness-of-fitverifythattheobservations

agreewith the assumption that the variableX obeys a normal distribution lawwith expectation = 10025 mm and standard deviation π = 1 mm if theinfluenceofround-offerrorsmaybeneglected

SOLUTION The theoretical distribution function F(x) is defined by theformula

Thestatisticaldistributionfunction (x)canbecalculatedbytheformula

ThecomputationsareperformedinTable50

TABLE50

Foreachvaluexiformingthedifferences andselectingfrom them the largest in absolute value according to Table 50 we findDq =00089

Computing

wefindthevalueofP(λ)fromTable25T

ThevalueofP(λ) is largeConsequently thedeviationsareinsignificantanditcanbeassumedthatthehypothesisthatthedataobeyanormaldistributionwithparameters = 10025 σ = 1 is valid however a large value of α leads todoubtsaboutthehighqualityofthesample

Following this solution one can solve Problems 435 438 4310 43124322and4323

Example434AccordingtothedataofExample432selectthedistributionlawbyusingaCharlier-Aseriesandtestbymeansofthechi-squaretestwhetherthe goodness-of-fit of the data with the resulting distribution law will beimprovedbyuseofthenormaldistribution

FromExample432wetaketheestimatesoftheexpectation andstandarddeviation

MoreoverusingthedataofTables47and48weestimate the thirdcentralmoment 3andthefourthcentralmoment 4oftherandomvariableX


TABLE51

Furthermorewe compute the estimates for the asymmetry and excess

bytheformulasUsingthefirstthreetermsofthedistributionfunctionfortheCharlier-Aseries

where

wefind

We now compute the values F(zi) and use tables 8T 10T for thedeterminationofthevaluesofФ(z)φ2(z)φ3(z)hereziare thecoordinateswithrespect to in units of the limits of the intervals The values of zi and thesubsequentcomputationsofF(zi)aregiveninTable52

TABLE52

ThetheoreticalprobabilitiespibasedonthedistributionlawdefinedbytheCharlier-Aseriesarecomputedbytheformula

Using thesevalues andnoting that we compute

(seeTable52)Thenumberofdegreesoffreedomisk=lndashrndash1=4sincethenumberof

classesisl=9(thelasttwointervalsbecauseoftheirsmallnumberareunitedintoasingle interval thenumberofparametersdeterminedon thebasisof the

data isr=4( )FromTable17T fork=4and =5615we

findthatαq= =0208The hypothesis on the goodness-of-fit of the experimental data with the

distributionlawF(z)specifiedbyaCharlier-AseriesisnotdisprovedHoweverthere are no reasons to assert that the goodness-of-fit is better than what isprovided by the normal distribution law mentioned in the assumption of theproblem

TnasimilarwayonecansolveProblems4326and4327

Example435Therearetwogroupsof60identical itemsproducedbytwomachines The data obtained from several measurements of some specificdimensionxoftheitemsaregiveninTable53

TABLE53

Test by means of the Kolmogorov test the hypothesis that both samplesbelongtoasinglepopulationiethatbothmachinesgivethesamedistributionofthedimensionxatasignificancelevelα=8percent

SOLUTION We divide the items into groups according to the increasingdimensionxandcomputethestatisticaldistributionfunctions 1(x)and 2(x)foreachgroup(seeTable54)

TABLE54

WefindthelargestabsolutevalueDn1n2ofthedifference 1(x)ndash 2(x)

Determining

whereinourcasen1=n2=6weobtainλ=09130UsingTable25TforλwehaveP(λ)=0375=αq

Thevalueofαq is large consequently the deviations are insignificant andthe hypothesis that both samples belong to the same main population is notcontradicted

Example 436 Six-hundred items are measured and for each item thedimensionsX andY are checkedThe results are given inTable55wherehijdenotesthenumberofitemswithdimensionsX=xiY=yj

TABLE55

ForXi=1ifthedimensionisunderestimatedi=2ifthedimensioniswithinthetolerancelimitsi=3ifthedimensionisoverestimatedforYj=123ifthe dimension is underestimatedwithin the tolerance limits or overestimatedrespectively

Testbyusingthechi-squaretestwhetherthedeviationsofdimensionsXandYfromadmissibledimensionsareindependentatasignificancelevelα=5percent

SOLUTIONWefindtheestimatesmijoftheexpectednumberofobservationsinwhichX=xiY=yjbystartingfromthehypothesisontheindependenceofX

andY

ThevaluesmijaregiveninTable56

TABLE56

Wecompute bytheformula

Thecomputations areperformed inTable57 inwhich thevalues of (hij ndashmij)2mijaregiven

TABLE57

Weget =2519Thenwedeterminethenumberofdegreesoffreedom

wherelisthenumberofgroupsaccordingtothedimensionXmisthenumberofgroupsaccordingtoYl=3m=3k=4UsingTable17Tfork=4and =

2519wefindαq= =0672

Thevalueofαqislargeconsequentlythehypothesisontheindependenceofthe deviations of the dimensions of an item according to the test ofX andYagainsttheadmissibledimensionsisnotdisproved

Problem4328canbesolvedsimilarly

PROBLEMS

431InTable58arelistedthenumbersmiofplotsofequalarea(025km2)ofthesouthernpartofLondonDuringtheSecondWorldWareachoftheplotswas hit by i buzz bombs Test with the aid of the chi-square test that theexperimental data agree with the Poisson distribution law

ifthesignificancelevelis6percent

TABLE58

432 For a thin layer of gold solution there is recorded the number ofparticlesofgold reaching the fieldofviewof amicroscopeduringequal timeintervalsTheresultsoftheseobservationsaregiveninTable59

TABLE59

Testwiththechi-squaretest thegoodness-of-fitof thePoissondistributionusethe5percentsignificancelevel

433Tenshotsarefiredfromarifleateachof100targetsandthehitsandthemissesarerecordedTheresultsappearinTable60

Testbyusingthechi-squaretestthattheprobabilitiesofhittingthetargetsarethesameforallshotsinotherwordstestthattheoutcomesobeyabinomialdistributionlawusethe10percentsignificancelevel

TABLE60

434 Seven coins are tossed simultaneously 1536 times and each time thenumberXofheadsisrecordedTable61liststhenumberofcasesinwhichthenumberofheadsisXi

TABLE61

Usingthechi-squaretesttestthehypothesisthattheexperimentaldataobeyabinomialdistributionlawAssumethattheprobabilityofoccurrenceofaheadis05foreachcoinThesignificancelevelshouldbe5percent

435Eachof100machinesproducesalotof40first-gradeandsecond-gradeitemsduringoneshiftSamplesof10 itemsfromeach lotareselectedandforeach sample the number of secondgrade items is recordedThe results of thetestsaregiveninTable62

TABLE62

The mi denote the numbers of samples with i second-grade items Thenumber of second-grade itemsproducedduring a longoperationperiodof theplantis30percent(p=030)ofallproduction

Test by using theKolmogorov test that the experimental results obey the

hypergeometric and binomial distribution laws if one uses the 5 per centsignificancelevel

For the quantity i distributed according to a hypergeometric law thereobtainstheformula

whereNisthenumberofitemsinthelotListhenumberofsecond-gradeitemsinthelotandnisthesamplesize

Forabinomialdistribution

436 Table 63 contains the deviations from a given dimension of thediametersofseveralcylindersproducedbyamachine

TABLE63

Test with the chi-square test the hypothesis that the observations obey anormaldistributionlawifthe5percentsignificancelevelisused

437Supposethat250numbersaregeneratedbysummingthedigitsoffive-digitnumbersselectedfromatableofrandomnumbersTheresultingsumsaredividedinto15intervalsasshowninTable64

TABLE64

Sums representing multiples of three are equally divided between twoadjacent intervals Using the chi-square test test whether the given statisticaldistribution obeys a normal distributionwhose parameters are the expectationandvariancedeterminedfromthedataifthesignificancelevelis5percent

438 Solve the preceding problem by using the Kolmogorov test Assume(becauseofthenarrownessoftheintervalinTable64)thatitispossibletotakeallelements ineach interval tobe thevalueat themidpointof the intervalToestablishthehypotheticalnormaldistributionlawconsiderthatanyvalue0to9fortheindividualdigitsofarandomfive-digitnumberhasprobabilityp=01

439Thedigits012hellip9amongthefirst800decimalsofthenumberπoccur749283798073777576and91timesrespectivelyUsingthechi-squaretesttestthehypothesisthatthesedataobeyauniformdistributionlawifthe10percentsignificancelevelisused

4310 Solve the preceding problem by using theKolmogorov test and byassumingthattheprobabilitythatanydigitappearsatanydecimalplaceis010

4311Froma tableof randomnumbers150 two-digitnumbers (00 is alsoconsideredatwo-digitnumber)areselectedTheresultsappearinTable65

TABLE65

Usingthechi-squaretestverifythehypothesisthattheobservationsobeyauniformdistributionlawfora5percentsignificancelevel

4312 Solve the preceding problem by applying the Kolmogorov testAssume (because of the narrowness of an interval in Table 65) that all theelementsinoneintervalmaybetakenequaltothemidpointoftheinterval

4313 The readings on the scale of a measuring instrument are estimatedapproximately in fractionsof onedivisionTheoretically anyvalueof the lastdigitisequallyprobablebutinsomecasescertaindigitsarefavoredoverothersInTable66200readingsofthelastdigitbetweenadjacentdivisionsofthescalearelistedUsingthechi-squaretestestablishwhetherthereisasystematicerrorinreadingsiewhethertheobservationsobeyauniformdistributionlawifthe

probabilityofappearanceofanydigitispi=010andthesignificancelevelis5percent

TABLE66

4314 The observed dailymean temperature of the air during 320 days isgiveninTable67

Establishwiththeaidofthechi-squaretestwhichofthetwodistributionsnormal orSimpson (triangular) agreeswith the data better if the significancelevelis3percent

TABLE67

4315InTable68therearelistedtheobservedtimeperiodsnecessarytofindandremovethefailureofacertainelectronicdevicetheseperiodsareexpressedinhourswithaprecisionofoneminute

TABLE68

Using the chindashsquare test test that the data obey a logarithmically normaldistribution inwhich x = log y obeys a normal distribution if the significancelevelis5percent

4316 The data of the Vorontsov-Velrsquoyaminov catalog the distribution ofdistancestoplanetarynebulaeareexhibitedinTable69whereXiisthedistance(inkiloparsecs)andmithenumberofcases(numberintheclass)

TABLE69

Using the chi-square test test the hypothesis that the data agree with thedistributionlawwhosedistributionfunctionF(|x|)hastheform

where and σ are the expectation and the standard deviation of the randomvariableXobeyinganormaldistribution lawandarerelated to theexpectationM[|X|] and the second moment m2 of the absolute |X| by the formulas

Herevistherootoftheequation

whereφ(v) andФ(v) are determined from tables 9T and 8T The significancelevelis5percent

4317 InTable70 the results of severalmeasurements of a quantityX aregiven

TABLE70

Usingthechi-squaretesttestthatthedataagreewiththenormaldistributionlaw and with the convolution of the normal and uniform distributions whoseparametersaretobedeterminedfromtheresultsofmeasurements

Remember that for the random variable X = Y + Z where Y and Z areindependent andY obeys a normal distribution lawwith zero expectation andvariance σ2 andZ obeys a uniform distribution law in the interval (α β) theprobability density ψ(x) is given by the expression

To determine the estimates of the parameters σ α β appearing in theformulaforψ(x) it is necessary toderive from thedata the estimates for theexpectation andthesecondandfourthcentralmoments 2and 4afterwhichthe estimates of σ α β are given by the equations

4318For602samplesthedistancer(inmicrons)ofthecenterofgravityofanitemtotheaxisofitsexteriorcylindricalsurfaceismeasuredwiththeaidofacontrolinstrumentTheresultsofthemeasurementsappearinTable71

TABLE71

Usingthechi-squaretestverifythatthedataobeyaRayleighdistribution

theestimateoftheparameterashouldbedeterminedintermsoftheestimate

fortheexpectationbytheformulaUsethe5percentsignificancelevel

4319Table72givestheresultsof228measurementsofthesensitivityXofatelevisionset(inmicrovolts)

TABLE72

Usingthechi-squaretestdeterminethebetterfitbetweenthenormalandtheMaxwelldistributionwhoseprobabilitydensityisdefinedbytheformula

AssumetheexpectationM[X]ofXandaarerelatedbytheformulaM[X]=x0+1596aForsimplicityselectasx0thesmallestobservedvalueofX

4320A lot of 200 light bulbs is tested for lifetimeT (in hours) and givesresultsasinTable73

TABLE73

Usingthechi-squaretesttestthatthedataobeyanexponentialdistributionlawwhoseprobabilitydensityisexpressedbytheformula

Thesignificancelevelshouldbetakenequalto5percentConsiderthefactthattheparameterλoftheexponentialdistributionlawis

relatedtotheexpectationoftherandomvariableTbytheformula

4321Alotof1000electronictubesistestedforlifetimeTable74givesthelifetimeintervals(titi+1)beforebreakdownsoccurandthecorrespondingsizesoftheclassestiareexpressedinhours

Using the chi-square test verify the hypothesis that the experimental dataagreewith theWeibull distribution lawThe distribution functionF(t) for this

lawisgivenbytheformula

where

Г(x)istheГ-functionTABLE74

Theparameters (theexpectedvalueofT)andmshouldbecomputedfromthedataTake intoaccount thatm is related to thestandarddeviationσby theformula

where

vm=σ isthecoefficientofvariationIn Table 32T there are given the values of bm and vm as functions ofm

KnowingvmwecanfindnandbmfromthistableThefollowingisasectionofthistable(Table75)

TABLE75

4322 The position of a pointM in the plane is defined by rectangularcoordinatesXandYAnexperimentconsistsofmeasuringtheangleφmadebytheradius-vectorofapointMwith they-axis (Figure36)Theresultsof1000measurements of φ rounded-off to the nearestmultiple of 15 degrees and thenumbersmiofappearancesofagivenvalueφiareshowninTable76

FIGURE36

TABLE76

If X and Y are independent normal variables with zero expectations andvariances equal to σ2 and (l4)σ2 respectively then z = tanφ must obey the

Cauchydistribution(thearctanlaw)Assumingthattherearenoerrorsinthemeasurementsofφandthattheround-offerrorsmaybediscountedtestbyusingtheKolmogorovtestthevalidityoftheprecedingassumptionsmadeaboutXandYifthesignificancelevelis5percent

4323Tochecktheprecisionofaspecialpendulumclockatrandomtimesonerecords theanglesmadebytheaxisof thependulumandtheverticalTheamplitudeofoscillationisconstantandequaltoα=15deg

Theresultsof1000suchmeasurementsrounded-offtothenearestmultipleof3degappearinTable77

TABLE77

Assuming that the round-off errors may be discounted test using theKolmogorovtestthehypothesisthatthedataagreewiththearcsinedistributionlawifthesignificancelevelis5percent

4324 To check the stability of a certain machine the following test isconductedeveryhourasampleof20itemsselectedatrandomismeasuredandusing the results of the measurements one computes in the ith sample theunbiasedestimateofthevariance Thevaluesof for47suchsamplesaregiveninTable78

TABLE78

Usingthechi-squaretesttestata5percentsignificancelevelthehypothesisofproportionalityof thevariances that is test the assumption that there isnodisorderwhichmeansthatthedispersionvarieswiththemeasureddimensionofanitemTakeintoaccountthefactthatif thishypothesisisvalid thequantity

obeys approximately a chi-square distribution law with (ni ndash 1) degrees offreedomwhere 2istheunbiasedestimateforthevarianceσ2oftheentiremain

populationandcanbecomputedbytheformulawhere ni = n = 20 is the number of items in each samplem = 47 is the

numberofsamplesand isthetotalnumberofitemsinallsamples

4325Therearem=40samplesofn=20itemseachandfortheithgroupthereisgivenasanestimatefortheexpectation iarandomlyselectedvaluexi1from the ith sample xi1 (for example the first in each sample) and for thevariance the unbiased estimate of the variance for the dimension x of anitemThevaluesof forthe40samplesappearinTable79

TABLE79

UsingtheKolmogorovtestverifyforthe10percentsignificancelevelthehypothesisthatthenormaldistributionobtainsforthedimensionx

Notethatinthiscase(fornne4)

where

obeyaStudentrsquosdistributionlawwithk=nndash2=18degreesoffreedomwherexijisarandomlyselectedvaluefromtheithsample(inourcasexi1)

4326The resultsof300measurementsof somequantityxare included inTable80

TABLE80

Usingthechi-squaretesttestthatthedataagreewiththenormaldistributionwhose parameter estimates should be computed from the experimental dataSmooth thedatawith theaidofadistributionspecifiedbyaCharlier-Aseriesand using the chi-square test verify that the data agree with the obtaineddistribution

4327ThemeasurementsoflightvelocitycintheMichelson-Pease-PearsonexperimentgavetheresultsshowninTable81Forbrevitythefirstthreedigitsofci(inkmsec)areomitted(299000)

TABLE81

Thefollowingestimatesfortheexpectedvalue andthestandarddeviation were obtained from the data

The chi-square test of the hypothesis that the data agree with a normaldistribution law with parameters and gives the value

thenumberofdegreesoffreedominthiscaseiskH=9

and small intervals are united The hypothesisshouldberejected

SmooththeobservationswiththedistributionlawspecifiedbyaCharlier-Aseries and test with the chi-square test that the experimental data obey theresultingdistributionlaw

4328 Two lots each containing 100 items aremeasured The number ofitems hij with normal underestimated and overestimated dimensions areexhibitedinTable82

TABLE82

Using the chi-square test determine whether the number of a lot and thecharacter of the dimensions of the items are independent at a 5 per centsignificancelevel

44 DATAPROCESSINGBYTHEMETHODOFLEASTSQUARES

BasicFormulas

Themethodof least squares is applied for finding estimates of parametersappearing in a functional dependence between variables whose values areexperimentallydetermined

Iftheexperimentgivesn+1pairsofvalues(xiyi)wherexiarethevaluesoftheargumentandyiarethevaluesofthefunctionthentheparametersoftheapproximating function F(x) are selected to minimize the sum

Iftheapproximatingfunctionisapolynomialthatis

thentheestimatesofitscoefficients karedeterminedfromasystemofm+1normal equations

where

Ifthevaluesxiaregivenwithouterrorsandthevaluesyiareindependentandequallyaccuratetheestimateforthevariance 2ofyi isgivenbytheformula

whereSministhevalueofScomputedundertheassumptionthatthecoefficientsof the polynomial F(x) = Qm(x) are replaced by their estimates that aredeterminedfromthesystemofnormalequations

Ifyiarenormallydistributed then themethodgiven isbest for finding theapproximatingfunctionF(x)

Theestimates akofthevariancesofthecoefficients kandthecovariances

are given by the formulas

whereMkj=ΔkjΔΔ=|dkj|isthedeterminantofthesystemofnormalequations

ofthe(m+l)storderΔkjisthecofactorofdkjinthedeterminantΔ

In solving the system of normal equations by the eliminationmethod thequantitiesMkjmayalsobeobtainedwithoutreplacingthevkbytheirnumericalvaluesThe linear combinationof thevkused to represent kwill have as thecoefficientofvjthedesirednumberMkj

Intheparticularcaseofalineardependencem=1wehave

In the case inwhich themeasurements are not equally accurate that isyihavedifferentvariances allthepreviousformulasremainvalidifSskandvk

arereplacedbywheretheldquoweightsrdquo ofyiare

A2isacoefficientofproportionalityIf theldquoweightsrdquopi are known the estimates of the variances of individual

measurementsyiarecomputedbytheformulaIfyiisobtainedbyaveragingniequallyaccurateresultsthentheldquoweightsrdquo

of the measurement yi is proportional to ni One may take = ni All theformulas remain unchanged except the one for in this case

The confidence intervals for the coefficients ak for any given confidencelevelhavetheform

whereγisdeterminedfromTable16TforStudentrsquosdistributionforthevaluesofαandk=nndashmdegreesoffreedom

Inthecaseofequallyaccuratemeasurementstheconfidenceintervalforthestandard deviation σ and the confidence level α are determined from theinequalitieswhereγ1andγ2arefoundfromTable19Tforachi-squaredistributionwithentryvalueαandkdegreesoffreedomForthesamepurposeonecanuseTable18T

inthiscasewhere and are determined from the equations

fork=nndashmdegreesoffreedomThe confidence limits form a strip containing the graph of the unknown

correctdependencey=F(x)withagivenconfidencelevelαtheyaredeterminedbytheinequalities

where (xi)istheestimateforthevarianceofydefinedbythedependencey=Qm(x)(itdependsontherandomvariablesrepresentedbytheestimatesofak)

Inthegeneralcasethecomputationof (x)isdifficultbecauseitrequirestheknowledgeof all thecovarianceskakal For a linear dependence (m = 1)

ThevalueofγisdeterminedfromTable16TforStudentrsquosdistributionfortheentryαandk=nndashmdegreesoffreedom

Inthecaseofequidistantvaluesxiof theargument thecomputationof theapproximating polynomial can be simplified by using the representation

where aretheorthogonalChebyshevpolynomials

xmaxxminarethemaximalandminimalvaluesofxi

Theestimatesforthevariancesofthecoefficientsbkaredeterminedbytheformula

ThevaluesoftheChebyshevpolynomialsmultipliedbyPkn(0)fork=1to5n=5to20xprime=01hellipnaregiveninTable30T

IfthecoefficientsbkarecomputedfromTable30Tthenforthecomputationof the polynomialsPk n(xprime) in the formula for Qm(x) it is also necessary toconsiderthecoefficientPkn(0)andtochoosetheordinatesofthesepolynomialsfrom the same tables or to multiply the value of the polynomial obtainedaccordingtotheprecedingformulabyPkn(0)

In some cases the approximating function is not a polynomial but canbereducedtoapolynomialbyachangeofvariablesExamplesofsuchchangearegiveninTable83

TABLE83

If y is a function of several arguments zi then to obtain the linearapproximatingfunction

correspondingtothevaluesyiandzki in (n+1)experiments it isnecessary tofind the solutions k of the system of normal equations

where

If the values zki are known without error and the measurements of yi areequally accurate the estimates of the variances of αk are determined by the

formulawhere 2 = Smin(n ndashm) andNk k is the ratio of the cofactor of a diagonalelementof thedeterminant(of thesystemofnormalequations) to thevalueofthedeterminantitselfInsolvingthesystemwithoutusingthedeterminantNkkwillbethesolutionsofthissystemifwereplaceallβkby1andtheotherβlbyzeros

Theroleofzkcanbeplayedbyanyfunctionsfk(x)ofsomeargumentxForexampleifthefunctionydefinedintheinterval(02π)isapproximatedbythe

trigonometricpolynomialthen for equidistant values xi the estimates for the coefficients λk and microk aredetermined by the Bessel formulas

For a complex functional dependence and a sufficiently small range ofvariationof theargumentszk thecomputationsare simplified if the function isexpanded inapower seriesofdeviationsofarguments from theirapproximatevalues(forexamplefromtheirmean)

If there are errors in xi and yi too and these variables obey a normaldistributiontheninthecaseoflineardependencetheestimate 1istherootofthequadraticequation

andtheestimate 0isgivenbytheformula

where arerespectivelythevariancesofthexiandtheyi

Ofthetworootsofthequadraticequationweselecttheonethatbetterfitstheconditionsoftheproblem


Example441Instudyingtheinfluenceoftemperaturetonthemotionωofachronometerthefollowingresultswereobtained(Table84)

TABLE84

If

holdswhere are the computedvaluesofωdetermine theestimates for thecoefficientsakand theestimatesfor thestandarddeviationsσofan individualmeasurementand akofthecoefficientsakEstablishtheconfidenceintervalsforakandforthestandarddeviationσcharacterizingtheprecisionofanindividualmeasurement for a confidence level α = 090 SOLUTION We determine thenormal equations for the coefficientsak andMkkTodecrease the sizes of the

coefficientsofthenormalequationsweintroducethevariable

andseektheapproximatingfunction

WethendeterminethecoefficientsofthenormalequationsskandvkasinthecomputationsinTable85

TABLE85

Weobtain

Thesystemofnormalequtionsbecomes

Solving this system by elimination andwithout substituting the numericalvaluesforvkweobtain

Substitutingthevaluesofvkwefind

Mk k are the coefficients of vk in each equation for that is

We compute the value Smin necessary for finding the estimates of thevariance of an individual yi and the variances of the coefficients k thecomputationsareinTable86

TABLE86

WeobtainSmin=0005223Furthermorewefind

Returningtotheargumenttweobtain

where

andthecorrespondingestimatesforthestandarddeviations ak

Wefindtheconfidenceintervalsforthecoefficientsakforaconfidencelevelα = 090UsingTable16T for thevaluesofα andk =n ndashm = 4 degrees offreedomwefind

Theconfidenceintervalsforak

become

Wefind theconfidence interval for thestandarddeviationσcharacterizingtheprecisionofanindividualmeasurement

whereγ1andγ2aredeterminedfromTable19Tfork=4α=090Wehaveγ1=0649γ2=237hence


Example 442 The results of several equally accurate measurements of aquantityyknowntobeafunctionofxaregiveninTable87

TABLE87

Selectafifth-degreepolynomialthatapproximatesthedependenceofyonxin the interval [027]Use (theorthogonal)ChebyshevpolynomialsEstimatethe precision of each individualmeasurement as characterized by the standarddeviationσandfindtheestimatesofthestandarddeviationsofthecoefficientsbkfortheChebyshevpolynomialsPkn(x)

SOLUTIONWemake the changeof variable z =x03 in order tomake theincreaseoftheargumentunityWecomputethequantitiesSkckbk(k=01hellip5) according to the formulas given in the introduction to this section The

tabulated values of the Chebyshev polynomials are taken from 30T ThecomputationsarelistedinTable88

TABLE88

The computations performed on a (keyboard) desk calculator withaccumulationoftheresultsgive

Fortheestimatesofthecoefficientsbkweget

Recall that ifoneuses the tabulatedvaluesof theChebyshevpolynomials theformulafortherequiredfifth-degreepolynomialhastheform

HoweverifoneusestheanalyticformulasforthecalculationoftheChebyshev

polynomials then thecoefficientsbk shouldbe replacedby wherePkn(0)isthetabulatedvalueofPkn(z)forz=0

Wecomputetheestimate 2

whereweusethetabulatedvaluesoftheChebyshevpolynomialsfromTable88forfindingthevalues iThecomputationofSminisindicatedinTable89

TABLE89

Weobtain

Nextaccordingtotheformula

wefind

Problems444446and4412canbesolvedbyfollowingthissolution

Example 443 The readings of an aneroid barometer A and a mercurybarometerBfordifferenttemperaturestaregiveninTable90

TABLE90

IfthedependenceofBontandAhastheform

find estimates of the coefficientsαk construct the confidence intervals for thecoefficientsαkandforthestandarddeviationσoftheerrorsinmeasuringBforaconfidencelevelα=090

SOLUTIONLetususethenotationsz0=1zi= tz2=760ndashAy=BndashAThentherequiredformulabecomes

TheinitialdataforthesenotationsarerepresentedinTable91

TABLE91

We determine the values

Wewritethesystemofnormalequationsbutfor thetimebeingwedonotreplaceβkbytheirnumericalvalues

Solvingthissystembyeliminationwefind

Setting the numerical values of βk in these expressions we find αk thecoefficients of βkin the expression for αkare the values of Nk k

FurthermorewefindSmin=08649(seeTable91)

We construct the confidence intervals for the coefficients αk and for thestandard deviation σ which determines the accuracy of an individualmeasurementbyusingStudentrsquosdistributionforαk(seeTable16T)andthechi-squaredistributionforσ(seeTable19T)

Thenumberofdegreesoffreedomisk=nndashm=7andtheconfidencelevelisα=090

Wefindγ=1897γ1=0705γ2=1797Theconfidenceintervalsforαk

become

andforthestandarddeviationσ

or

Example444Table92containsthevaluesxiyiandtheldquoweightsrdquo thatdeterminetheaccuracyinmeasuringyiforagivenvaluexi

TABLE92

Ifyisasecond-degreepolynomialinx

find the estimates for the variances of individual measurements of yi and thevariancesofthecoefficientsak(k=012)Constructtheconfidencelimitsfortheunknowntruerelationy=F(x)ataconfidencelevelα=090

SOLUTIONWecompute thequantities and for the systemofnormalequationsbutconsidertheldquoweightrdquoofeachmeasurementThecomputationsaregiveninTable93

TABLE93

Weobtain

Wewritethesystemofnormalequations

We find the numerical values of the determinant Δ of the system and thecofactors δkj of the elements dkj = of this determinant

Wecomputetheestimatesofthecoefficientsak

andget

WefindSminbyperformingthecomputationsgiveninTable94

TABLE94

Wecompute the estimatesof thevariancesof individualmeasurementsbytheformula

andobtain

Theestimatesofthevariancesofthecoefficientsakandtheircovariancesare

givenbytheformulas

Wehave

We calculate the estimate of the variance of by the formula

orby

Thevalues forallxiarecalculatedinTable95Weconstructtheconfidencelimitsfortheunknowntruerelationy=F(x)

whereγisdeterminedfromTable16Tforα=090andk=nndashm=6degreesoffreedom

TheconfidencelimitsforyarecomputedasinTable95SimilarlyonecansolveProblems447448and4411

TABLE95

Example445Thevaluesof theelectricresistanceofmolybdenumdependontemperatureTdegKasshowninTable96

TABLE96

IfρislinearlydependentonT

determinethecoefficientsa0anda1bythemethodofleastsquaresTheerrorsinmeasurementsofρandTarespecifiedbythestandarddeviations =08andσT=15deg respectivelyFind themaximaldeviationof thecalculatedvalueofρfromtheexperimentalone

SOLUTIONWecalculatethequantitiesSkrk(k=12)v1asshowninTable97

TABLE97

weobtion

Wewritethequadraticequationforthecoefficient 1

whichafterthesubstitutionofthenumericalvaluesbecomes

Solvingthisequationwefindtwovaluesfor 1

Obviously the negative root 12 is extraneous since the data contained inTable 97 show that when T increases ρ increases Consequently

Wedeterminethecoefficient 0bytheformula

Wecalculatethevaluesof 0inTable97

where arethecomputedvaluesofthequantity

FromthedataofTable97wefindthat|εmax|=028OnecansolveProblem4415similarly

PROBLEMS

441TheresultsofseveralequallyaccuratemeasurementsofthedepthhofpenetrationofabodyintoabarrierfordifferentvaluesofitsspecificenergyE(thatistheenergyperunitarea)aregiveninTable98

TABLE98

Selectalinearcombinationoftheform

Determine the estimates of the variances of the coefficients ak and theestimate 2 of the variance determining the accuracy of an individualmeasurement

442SolvetheprecedingproblembyshiftingtheoriginofEtothearithmeticmeanofEandtheoriginofhtoapointclosetotheexpectationofhandtherebysimplifythecomputations

443Theheighthofabodyinfreefallattimetisdeterminedbytheformula

wherea0istheheightatt=0a1istheinitialvelocityofthebodyanda2ishalftheaccelerationofgravityg

Determine the estimates of the coefficients a0 a1 a2 and estimate theaccuracyofdeterminationoftheaccelerationofgravitybytheindicatedmethodby using a series of equally accurate measurements whose results appear inTable99

TABLE99

444 Solve the preceding problem by using (the orthgonal) Chebyshevpolynomials

445SeveralequallyaccuratemeasurementsofaquantityyatequallyspacedvaluesoftheargumentxgivetheresultsappearinginTable100

TABLE100

Ifyisquiteaccuratelyapproximatedbythesecond-degreepolynomial

determine the estimates of the coefficients k the variance of an individual

measurement 2andthevariances ofthecoefficients k446 The amount of wear of a cutter is determined by its thickness (in

millimeters)asafunctionofoperatingtimet(inhours)TheresultsaregiveninTable101

TABLE101

Using(theorthogonal)Chebyshevpolynomialsexpressybothasafirst-andthenasathird-degreepolynomialoftConsideringthattheresultsarevalidinbothcasesestimatethemagnitudeofthevarianceofanindividualmeasurementand construct the confidence intervals for the standard deviation σ for aconfidencelevelα=090

447Thevalueof thecompressionofasteelbarxiundera loadyiand the

valuesofthevariances whichdeterminetheaccuracyinmeasurementsofyiaregiveninTable102

TABLE102

Findthelineardependence

associated with Hookersquos law Construct the confidence intervals for thecoefficientsak(k=01)andalsotheconfidencelimitsfortheunknowncorrectvalueoftheloadforxrangingfrom5to60microiftheconfidencelevelisα=090

The ldquoweightsrdquo of themeasurements corresponding to each value xi of the

compressionaretakeninverselyproportionalto 448Table103containstheaveragevaluesofyicorrespondingtothevalues

xioftheargumentandalsothenumberniofmeasurementsofyfoxx=xi

TABLE103

Construct the approximating second-degree polynomial and determine theestimatesofthestandarddeviations ofthecoefficients k

449 The net cost (in dollars) of one copy of a book as a function of thenumber(inthousandsofcopies)inagivenprintingischaracterizedbythedataaccumulatedbythepublisheroverseveralyears(Table104)

TABLE104

Selectthecoefficientsforahyperbolicdependenceoftheform

andconstructtheconfidenceintervalsforthecoefficients(k=01)andalsoforthequantityyfordifferentvaluesofxiiftheconfidencelevelisα=090

4410 A condenser is initially charged to a voltage U after which it isdischarged through a resistance The voltageU is rounded-off to the nearestmultipleof5voltsatdifferenttimesTheresultsofseveralmeasurementsappearinTable105

TABLE105

ItisknownthatthedependenceofUonthastheform

SelectthecoefficientsU0andaandconstructtheconfidenceintervalsforU0andaforaconfidencelevelα=090

4411The following data obtained from an aerodynamical test of amodelairplane(seeTable106)expressthedependenceoftheangleofinclinationδB(oftheelevatorensuringarectilinearhorizontal flight)on thevelocityυof theair

stream

TABLE106

Findtheestimatesofthecoefficientsa0anda1andtheirstandarddeviationsThenidenotethenumberofmeasurementsforagivenvalueofthevelocityυi

4412 The results of several measurements of the dimension x of a lot ofitems are divided into intervals and the frequencies in Table 107 arecomputedforthem

TABLE107

If thevaluesof refer to themidpointsof the intervalsxi select by themethod of least squares the parameters for the relation

that approximates the experimental distribution Apply (the orthogonal)ChebyshevpolynomialsTestwhethertheresultingdependenceobeysanormaldistribution law for x that is whether the following equation holds

4413 Table 108 contains the measured values of some quantity y as afunctionoftimet(fora20hourperiod)

TABLE108

If

determinetheestimatesoftheparametersaandφFindthemaximaldeviationofthemeasuredquantityyfromtheapproximatingfunction

HintFirstchoosetheapproximatevalueφprimeandrepresentyintheform

where

4414Table109containstheexperimentaldataforthevaluesofafunctiony=f(x)withperiod2π

TABLE109

Findtherepresentationofthisfunctionbythepolynomial

and themaximaldeviationof themeasuredquantityy from theapproximatingfunction

4415Table110containsthelevelsxandyofthewaterinariveratpointsAandBrespectively(Bis50kmdownstreamfromA)Theselevelsaremeasuredatnoonduringthefirst15daysofApril

TABLE110

Iftherelation

holds determine the estimates of the coefficients 0 and 1 and themaximaldeviation yi from the calculated values i if it is known that the errors inmeasurementsofxandyarecharacterizedbystandarddeviationsσx=σy=05m

45 STATISTICALMETHODSFORQUALITYCONTROL

BasicFormulas

Qualitycontrolmethodspermitus toregulateproductqualityby testingAlot of items is sampled according to a scheme guaranteed to reject a good lotwith probability α (ldquosupplierrsquos riskrdquo) and to accept a defective lot withprobabilityβ(ldquoconsumerrsquosriskrdquo)

Alot isconsideredgoodif theparameter thatcharacterizes itsqualitydoesnotexceedacertainlimitingvalueanddefectiveifthisparameterhasavaluenotsmallerthananotherlimitingvalueThisqualityparametercanbethenumberlofdefectiveitemsinthelot(withthelimitsl0andl1gtl0)theaveragevalueofξorλ (with the limitsξandξ1gtξ0orλ0 andλ1gtλ0) or (for the homogeneitycontroloftheproduction)thevarianceoftheparameterinthelot(withthelimits

and )Inthecaseinwhichthequalityofalotimproveswiththeincreaseoftheparameterthecorrespondinginequalitiesarereversed

TherearedifferentmethodsofcontrolsinglesamplingdoublesamplingandsequentialanalysisThedeterminationofthesizeofthesampleandthecriteriaof acceptance or rejection of a lot according to given values of α and βconstitutesplanning

In the case of singlesampling one determines the sample sizen0 and theacceptancenumbervifthevalueofthecontrolledparameterislevinthesamplethenthelotisacceptedifitisgtvthenthelotisrejected

Ifonecontrolsthenumber(proportion)ofdefectiveitemsinasampleofsizen0thetotalnumberofdefectiveitemsinthelotbeingLandthesizeofthelot

beingNthenwherethevalues canbetakenfromTable1TorcomputedwiththeaidofTable2T

Forn0le01Nitispossibletopassapproximatelytoabinomialdistributionlaw

wherep0=l0Np1=l1NandthevaluesofP(pnd)canbetakenfromTable4TorcomputedwiththeaidofTables2Tand3T

Moreoverifp0lt01p1lt01thenlettinga0=n0p0a1=n0p1 (passing tothe Poisson distribution law) we obtain

where

aregiveninTable7Tandtheprobabilities canbeobtainedfromTable17Tfork=2(v+1)degreesoffreedom

If50len0le01Nn0p0ge4thenonemayusethemoreconvenientformulas

whereФ(z)istheLaplacefunction(seeTable8T)Ifonecontrols theaveragevalue of theparameter

in a sample and the value of the parameter xi of one item obeys a normaldistribution with known variance σ2 then

Forξ0 gtξ1 the lot is accepted if gev it is rejected if ltv and in theformulasforαandβtheminussignisreplacedbyplussign

Ifthecontrolledparameterhastheprobabilitydensity

then

where and the probabilityis determined by Table 17T for k = n0 degrees of freedom If n0 gt 15 then

approximatelyIfonecontrolstheproducthomogeneityandthequalityparameterisnormal

then

where if theexpectation of the parameter is known or

if is unknown and theprobabilities are calculated fromTable22T for k =n0 degrees of freedom if x is known and for k =n0 ndash 1 if isunknown

Inthecaseofadoublesamplingonedeterminesthesizesn1ofthefirstandn2 of the second samples and the acceptance numbers v1v2 v3 (usually v1 lt[n1(n1+n2)]v3ltv2)Ifinthefirstsamplethecontrolledparameterislev1thenthelotisacceptedifthecontrolledparameterisgtv2thenthelotisrejectedinthe other cases the second sample is taken If the value of the controlledparameterfoundforthesampleofsize(n1+n2)islev3thenthelotisacceptedandotherwiseitisrejected

Ifonecontrolsbythenumberofdefectiveitemsinasamplethen

As in the case of single sampling in the presence of certain relationsbetweenthenumbersn1n2Nl0l1anapproximatepassageispossiblefromahypergeometricdistributiontoabinomialnormalorPoissondistributionlaw

Ifonecontrolsbytheaveragevalue oftheparameterinasamplethenforanormaldistributionoftheparameterofoneitemwithgivenvarianceσ2intheparticular case when n1 = n2 = n v1 = v3 = v v2 = infin we have

where

Forξ0 gtξ1 the inequality signs appearing in the conditions of acceptance andrejection are reversed and in the formulas for p1 p2 p3 p4 the plus signappearinginfrontofthesecondtermisreplacedbyaminussign

Ifonecontrolsby and theprobabilitydensityof theparameterX foroneitem is exponential f(x) = λendashλx n1 = n2 = n v1 = v3 = v v2 = infin then

where

and the probabilities arecomputedaccordingtoTable17Tfork=2ndegreesoffreedom(forp1andp3)andk=4n(forp2andp4)

If one controls the homogeneity of the production when the controlledparameter is normally distributed n1 = n2 = n v1 = v3 = v v2 = infin then

wherep1p2p3p4aredeterminedfromTable22Tforq=q0forp1andp2q=q1forp3andp4foraknown k=nforp1andp3k=2nforp2andp4foranunknown k=nndash1forp1andp3k=2nforp2andp4

In the sequentialWaldanalysis for a variable sample sizen anda randomvalueof the controlled parameter in the sample the likelihood coefficient γ iscomputed and the control lasts until γ leaves the limits of the interval (BA)whereB=β(1ndashα)A=(1ndashβ)αifγleBthenthelotisacceptedifγgeAthelotisrejectedandforBltγltAthetestscontinue

Ifonecontrolsbymeansofmdefectiveitemsinasamplethen

Fornle01Naformulavalidforabinomialdistributionisuseful

where

Inthiscasethelotisacceptedifmleh1+nh3thelotisrejectedifmgeh2+nh3and the tests continue if h1 + nh3 lt m lt h2 + nh3 where

InFigure37thestripIIgivestherangeofvaluesfornandmforwhichthetests are continued I being the acceptance range and III being the rejectionrange

Ifnle01Np1lt01then

wherea0=np0a1=np1Forthemostparttheconditionsforsequentialcontroland the graphical method remain unchanged but in the present case

Ifthebinomialdistributionlawisacceptabletheexpectationofthesamplesizeisdeterminedbytheformulas

FIGURE37

The expectation of the sample size becomes maximal when the number ofdefectiveitemsinthelotisl=Nh3

Ifonecontrolsbytheaveragevalue oftheparameterinthesampleandtheparameterofoneitemisanormalrandomvariablewithknownvarianceσ2then

Thelotisacceptedif thelotisrejectedif

and the tests are continued if where

ThemethodofcontrolinthepresentcasecanalsobegraphicallyrepresentedasinFigure37ifn isusedinplaceofmonthey-axisForξ0gtξ1weshallhaveh1 gt 0 h2 lt 0 and the inequalities in the acceptance and rejection conditionschangetheirsigns

Theexpectednumberoftestsisdeterminedbytheformulas

If the parameter of an individual itemhas the probability density f(x) = λendashλxthen

Thelotisacceptedif itisrejectedif andthe tests are continued if where

The graphical representation of the method of control differs from thatrepresented in Figure 37 only because in the present case I represents therejectionregionandIIIrepresentstheacceptanceregionTheexpectednumber

of tests is computed by the formulas

Iftheproductionischeckedforhomogeneity(normaldistributionlaw)then

The lot is accepted (for a known ) if it is rejected if and the tests are continued if

where

ThegraphicalrepresentationisanalogoustoFigure37withthevaluesof onthey-axis

If is unknown then whenever n appears in the formulas it should bereplacedby(nndash1)

Theexpectednumbersoftestsare

Ifthetotalnumberofdefectsoftheitemsbelongingtothesampleischeckedand thenumberofdefects ofone itemobeys aPoisson lawwithparameterathenalltheprecedingformulasareapplicableforthePoissondistributionifwereplace

mbyn p0andp1bya0andala0anda1byna0andnal by2na0andby2nalwherenisthesizeofthesample

Fornge50nage4itispossibletopasstoanormaldistribution

Todeterminetheprobabilitythatthenumberoftestsisnltnginasequentialanalysis when α β or β α one may apply Waldrsquos distribution

wherey is theratioofthenumberof tests(n) to theexpectationofn forsome

value of the control parameter of the and theparametercofWaldrsquosdistributionisdeterminedbythefollowingformulas(a)forabinomialdistributionoftheproportionofthedefectiveproduct

(b)foranormaldistributionoftheproductparameter

(c)foranexponentialdistributionoftheproductparameter

where

A special case of control by the number of defective products arises inreliabilitytestsofdurationtwherethetimeofreliableoperationisassumedtoobeyanexponentialdistributionlawInthiscasetheprobabilitypthatanitemfails during time t is given by the formula p = 1 ndash endashλt All the formulas ofcontrol for the proportion of defective products in the case of a binomialdistributionremainvalidifonereplacesp0by1ndashendashλ0tp1by1ndashendashλ1tIfλtlt01then it is possible to pass to a Poisson distribution if in the corresponding

formulas one replaces a0 by nλ0t a1 by by2nλ1t

Thesequentialanalysisdiffersinthepresentcasebecauseforafixednumbern0oftesteditemsthetestingtimetisrandomThelotisacceptediftget1+mt3rejectediftget2+mt3andthetestsarecontinuedift1+mt3gttgtt2+mt3where

andmisthenumberoffailuresduringtimetToplotthegraphonerepresentsmonthex-axisandtonthey-axis

The expectation of the testing time T for λt lt 01 is determined by theformulas

where tH isanumberchosen tosimplify thecomputationsandp0=λ0tHp1 =λ1tH

TodeterminetheprobabilitythatthetestingtimeTlttgifλ βorβ λoneappliesWaldrsquosdistributioninwhichoneshouldsety=tM[T|λ]andfindtheparameterc by the formula valid for a binomial distribution for the precedingchosenvalueoftH


Example451AlotofN=40itemsisconsideredasfirstgradeifitcontainsatmostl0=8defectiveitemsIfthenumberofdefectiveitemsexceedsl1=20thenthelotisreturnedforrepairs

(a)Computeαandβbyasinglesamplingofsizen0=10iftheacceptancenumberisv=3

(b)findαandβforadoublesamplingforwhichn1=n2=5v1=0v2=2v3 = 3 (c) compare the efficiency of planning by the methods of single anddouble samplings according to the average number of items tested in 100identicallots

(d) construct the sequential sampling plan for α and β obtained in (a)determinenminforthelotswithL=0andL=N

SOLUTION(a)Wecomputeαandβbytheformulas

UsingTable1Tfor wefind

(b)Wecomputeαandβbytheformulas

andobtain

(c)Theprobabilitythatafirst-gradelotinthecaseofdoublesamplingwillbeacceptedafterthefirstsamplingoffiveitemsis

Theexpectationofthenumberoflotsacceptedafterthefirstsamplingfromatotalnumberof100lotsis

fortheremaining694lotsasecondsamplingisnecessaryTheaveragenumberofitemsusedindoublesamplingis

Inthemethodofsinglesamplingthenumberofitemsusedis

In comparing the efficiencyof the controlmethodswehaveneglected thedifferences between the values of α and β obtained by single and doublesampling

(d) For α = 0089 and β = 0136 the plan of sequential analysis is thefollowing

Todeterminenminwhenalltheitemsofthelotarenondefectivewecomputethesuccessivevaluesoflogγ(n0)bytheformulas

Wehave

Sincetheinequalitylogγ(n0)ltltlogBissatisfiedonlyifnge8itfollowsthatnmin=8

Foralotconsistingofdefectiveitemsn=mWefindlogγ(l1)=03979Forsuccessivevaluesofnwemakeuseoftheformula

We obtain log γ(2 2) = 08316 log γ(3 3) = 13087 gt log A = 0987consequentlyinthiscasenmin=3


Example452Alargelotoftubes(Ngt10000)ischeckedIftheproportionofdefectivetubesisplep0=002thelotisconsideredgoodifpgep1=010thelotisconsidereddefectiveUsingthebinomialandPoissondistributionlaws(confirmtheirapplicability)(a)computeαandβ forasinglesampling(singlecontrol)ifn=47v=2

(b)computeαandβforadoublesampling(doublecontrol)takingn1=n2=25v1=0v2=2v3=2 (c) compare theefficiencyof the singleanddoublecontrolsbythenumberofitemstestedper100lots

(d)constructtheplanofsequentialcontrolplotthegraphanddeterminenminforthelotwithp=0p=1computetheexpectationforthenumberoftestsinthecaseofsequentialcontrol

SOLUTION(a)Inthecaseofbinomialdistribution

UsingTable4Tforthebinomialdistributionfunctionandinterpolatingbetweenn=40andn=50wegetα=00686β=01350

InthecaseofaPoissondistributionlawcomputingα0=n0p0=094a1=n0p1 = 47 we obtain

UsingTable7TwhichcontainsthetotalprobabilitiesforaPoissondistributionwefind(interpolatingwithrespecttoa)

(b)ForabinomialdistributionlawusingTable1Tand4Twefind

In the case of a Poisson distribution law using Tables 6T and 7T andcomputing α01 = 05 a02 = 05 a11 = 25 a21 = 25 we obtain

The essential difference between the values of βcomputed with the aid ofbinomialandPoissondistributionsisexplainedbythelargevalueofp1=010

(c) The probability of acceptance of a good lot (p le 002) after the firstsamplinginthecaseofdoublecontrol(wecomparetheresultsofthebinomialdistribution) is

Theaveragenumberofgoodlotsacceptedafterthefirstsamplingfromthe

totalnumberof100lotsis

Fortheremaining3965lotsasecondsamplingwillbenecessaryTheaverageexpenditureintubesforadoublecontrolof100lotsisequalto

Inadefectivelottheprobabilityofrejectionafterthefirstsamplinginthecaseofdoublecontrolis

Theaveragenumberoflotsrejectedafterthefirstsamplingfromatotalof100lotsis

Fortheremaining5371lotsasecondsamplingwillbenecessaryTheaverageexpenditureintubesforadoublecontrolof100lotswillbe

Forasinglecontrolinallcases

willbeconsumed(d) Forα = 00686 β = 01350 for a sequential control using a binomial

distributionweget

Furthermoreh1=ndash1140h2=1496h3=00503(Figure38)Wefindnminforagood lot for p = 0

foradefectivelotwhenp=1

Wedeterminetheaveragenumbersoftestsfordifferentp

FIGURE38

Problems452to455457458and4510canbesolvedbyfollowingthissolution

Example 453 A large lot of resistors for which the time of reliableoperationobeysanexponentialdistributionissubjectedtoreliabilitytestsIfthefailureparameterλleλ0=2middot10ndash6hoursndash1thelotisconsideredgoodifλgeλ1=1middot10ndash5hoursndash1thelotisconsidereddefectiveAssumingthatλt0lt01wheret0is a fixed testing time for each item in a sampleof sizen0 determine forα =0005β=008thevalueofn0Usethemethodofsinglesamplingfordifferentt0findvwiththeconditionthatt0=1000hoursandalsoconstructtheplanofsequentialcontrolinthecasen=n0fort0=1000hoursComputetminforagoodlotandadefectiveoneandM[T|λ]P(tlt1000)P(tlt500)

SOLUTION The size n0 of the sample and the acceptance number v aredeterminedbynotingthatλt0lt01whichpermitsuseofthePoissondistributionand furthermore permits passing from a Poisson distribution to a chi-squaredistributionWecomputethequotientλ0λ1=02NextfromTable18Twefind

thevalues for theentryquantities =1ndashα=0995andkfor andkBythemethodofsamplingweestablishthatfork=15

fork=16

Interpolatingwithrespectto =02wefindk=1563 =487=2399Wecomputev=(k2)ndash1=6815wetakev=62n0λ0t0=487

hence it follows thatn0t0=4872middot0000002=1218middot10ndash6Theconditionλt0lt01 leads to

Taking different values t0 lt 10000we obtain the corresponding values ofn0giveninTable111

TABLE111

WecomputeBAt1t2forthemethodofsequentialanalysisB=008041InB=ndash25211A=184InA=52161Takingn0=1218wehavet1=2587hours

t2=ndash5353hours

t3=1652hours(Figure39)Theminimaltestingtimeinthecasewhenm=0foragoodlotistmin=2587hoursforadefectivelottmin=ndash5353+1652mgt0m=324asymp4form=4tmin=1255hoursIffortlt1255hoursmge4thenthelotisrejected

Tocompute the average testing time forn =n0 = 1218we take tH = t0 =1000hoursThen

Furthermorewefind

FIGURE39

thenwecompute

Wefindtheprobabilitythatthetestingtimeforafixednumberofitemsn=n0=1218islessthan1000hoursand500hoursThereforefortH=1000hourswecompute the value of the parameter c ofWaldrsquos distribution and the value of

withtheconditionthatp0=λ0t0=0002p1=λ1t0=001Takingp=p0sinceαβweobtainc=237y=1000415=2406Wefindthat(seeTable26T)

Forγ=05wehave

OnecansolveProblem459similarly

Example454Thequalityofthedisksproducedonaflat-grindingmachineisdeterminedbythenumberofspotsonadiskIftheaveragenumberofspotsper10disksisatmostonethenthedisksareconsideredtobeofgoodqualityiftheaveragenumberisgreaterthanfivethenthedisksaredefectiveAsampleof40disks isselectedfroma large lot (Ngt1000)Assuming that thenumberofspotsonadiskobeysaPoissondistributionlaw(a)determineαandβforv=9

(b)fortheseαandβconstruct theplanofsequentialcontrolcomputenmingood lotandadefectiveoneand find thevaluesofM[n|a] (c) test aconcretesamplewhosedataappearinTable112bythemethodsofsingleandsequentialcontrol

TABLE112

SOLUTION(a)UsingthePoissondistributionwehavea0=01a1=05na0=4na1 = 20UsingTable7T for the total probabilities of xn occurrences ofspots on disks in the sample we find

(b)Forα=00081β=00050 thecharacteristicsof thesequentialcontrol

(Figure40)are

Wecomputenmin

Theaveragenumberoftestsinthecaseofsequentialcontrolis

(c)Inasamplewithn0=40itturnsoutthatxn=7ltv=9consequentlythelotisacceptedApplyingthemethodofsequentialcontrol(seeFigure40)forn=30weobtainthatthepointwithcoordinates(nm)liesbelowthelowerlinethat is the lot should be accepted Indeed


Example 455 The quality of punchings made by a horizontal forgingmachine is determined by the dispersion of their heightsX known to obey a

FIGURE40

normaldistribution lawwith expectation =32mm (nominal dimension) Ifthestandarddeviationσleσ0=018mmthelotisconsideredgoodifσgeσ1=030mmthelotisdefectiveFindαandβforthemethodofsinglesamplingifn0=39andv=022mmUse the resultingvalues forα andβ to construct acontrolplanbythemethodofsequentialanalysisComputenmin foragoodlotandadefectiveoneandfindM[n|σ]

SOLUTIONWecomputeαandβbytheformulas

fork=n0=39q0=vσ=1221q1=vσ1=0733InterpolatingaccordingtoTable 22T for the chi-square distribution we find

WefindthevaluesofBAh1h2h3forthemethodofsequentialanalysis

WefindnminForthepoorestamongthegoodlots

Forthebestamongthedefectivelots nmin =h2+nminh3nmin=93asymp10

WecomputetheaveragenumbersoftestsM[n|σ]fordifferentσ

InasimilarmanneronecansolveProblem4512

Example456ThemaximalpressureX inapowderchamberofarocketisnormally distributed with standard deviation σ = 10 kgcm2 The rocket isconsideredgoodifXle=ξ0=100kgcm2ifXgeξ1=105kgcm2therocketisreturned to the plant for adjustmentGiven the valuesα = 010 andβ = 001constructtheplansforsinglecontrol(n0v)andsequentialcontrolcomputetheprobabilities P(n lt n0) and P(n lt(frac12)n0) that for the sequential control theaveragenumberoftestswillbelessthann0and(frac12)n0respectively

SOLUTIONTocomputethesamplesizen0andtheacceptancenumbervforasinglecontrolweusetheformulas

SubstitutingthevaluesforαandβandusingTable8TfortheLaplacefunctionwefind

henceitfollowsthatn0=52v=1018kgcm2ForthesequentialcontrolwefindthatB=00111lnB=ndash4500A=99

lnA=2293h1=ndash90h2=4586h3=1025WedeterminehminForthepoorestamongthegoodlotswhen =ξ0=100

forthebestamongthedefectivelotswhen =ξ1=105

TheaveragenumberofobservationsM[n|ξ]isequalto

Todetermine theprobabilityP(nlt52) sinceα β for =ξ1=105we

computeFromTable26TforWaldrsquosdistributionlawwefindthat

ByfollowingthissolutiononecansolveProblem4513

Example 457 The average time of operation of identical electron tubesrepresentstget0=1282hoursforagoodlotandtlet1=708hoursforadefectiveone It is known that the time T of reliable operation obeys an exponentialdistributionlawwiththeprobabilitydensitywheretheparameterλistheintensityoffailuresthatistheinverseofthemeantimeofoperationofatubeinhours

Determineforα=0001andβ=001 thesizen0of thesinglesampleandthe acceptance number v construct the sequential control plan and find nminM[n|λP(nltn0)P(nlt(12)n0)

SOLUTIONAssumingthatn0gt15(sinceαandβaresmall)wereplace thechi-square distribution which the quantity 2λn0 obeys by a normal

distributioniewesetsincethenumberofdegreesoffreedomisk=2nWeobtaintheequations

henceitfollowsfromTable8Tthat

or since λ0 = 1t0 = 000078 λ1 = 1t1 =

0001413Ifwesolvethissystemofequationsweobtain

Sincen0gt15theuseofanormaldistributionispermissibleForthesequentialcontrolwefindthat

WedeterminenminFor thepoorestamong thegood lots = t0=1282hoursnmin=211asymp22forthebestamongthedefectivelots =t1=708hoursnmin=474asymp48

Wefindtheaveragenumbersoftestsfordifferentλ

Sinceα βwedetermineK=|lnB|=4604andthentheparametercofWaldrsquosdistributionc=1525furthermorewefindy01=100207=482y02=241

FromTable26Tfory01(y02)andcwehave

SimilarlyProblem4514canbesolved

PROBLEMS

451Rodsinlotsof100arecheckedfortheirqualityIfalotcontainsLlel0=4defectiveitemsthelotisacceptedifLgel1=28thelotisrejectedFindαandβforthemethodofsinglesamplingifn0=22v=2andforthemethodofdouble sampling for n1 = n2 = 15 v1 = 0 v2 = 3 v3 = 3 compare theirefficiencies according to the average number of tests construct the sequentialanalysis plan and compute the minimal number of tests for a good lot and adefective one in the case of sequential control Use the values of α and βobtainedbythemethodofsinglesampling

452Intheproductionoflargelotsofballbearingsalotisconsideredgoodifthenumberofdefectiveitemsdoesnotexceed15percentanddefectiveifitexceeds5percentConstructandcompare theefficiencyof theplanof singlecontrolforwhichthesamplesizen0=410andacceptancenumberv=10andtheplanofdoublecontrolforwhichn1=n2=220v1=2v2=7v3=11

Construct thesequentialcontrolplanwithαandβas foundfor theplanofsingle control Compare the efficiencies of all threemethods according to theaveragenumberoftestsandcomputenminforagoodlotandadefectiveoneforsequentialcontrol

453A large lot of punched items is considered good if the proportion ofdefectiveitemsplep0=010anddefectiveifpgep1=020Findαandβforthecontrolbysinglesamplingusesamplesizen0=300andacceptancenumberv=45FortheresultingvaluesofαandβconstructthecontrolplanbythemethodofsequentialanalysisandcomputenminforagoodlotandadefectiveonefindM[n|p]andP(nltn0)P(nlt(12)n0)

HintPasstothenormaldistribution454Foralargelotofitemsconstructtheplanofsinglecontrol(n0v) that

guarantees(a)asupplierrsquosriskof1percentandaconsumerrsquosriskof2percentifthelotisacceptedwhentheproportionofdefectiveitemsisplep0=010andrejectedwhenpgep1=020(usethenormaldistribution)(b)α=020β=010

for the same p0 and p1 applied to a Poisson distribution law Construct thecorresponding plans of sequential control and find the expectations for thenumberoftests

455Forα=005andβ=010constructtheplansofsingleandsequentialcontrolforqualitytestsoflargelotsofrivetsTherivetsareconsidereddefectiveiftheirdiameterXgt13575mmAlotisacceptediftheproportionofdefectiverivets isp lep0 = 003 and rejected ifP geP1 = 008Compute for a Poissondistributionthesizen0of thesinglesampleand theacceptancenumbervForthesameα andβ construct theplanof sequential control computenmin for agoodlotandadefectiveoneandfindtheaveragenumberoftestsM[n|p]inasequentialcontrol

456RivetswithdiameterXgt13575mmareconsidereddefectiveAtmost5percentofthelotswhoseproportionofdefectiveitemsispltp0=003mayberejectedandatmost10percentoflotswhoseproportionofdefectiveitemsispge p1 = 008 may be accepted Assuming that the random variableX obeys anormal distributionwhose estimates of the expectation and variance aredeterminedonthebasisofsampledatafindthegeneralformulasforthesizen0of the single sample in dimension control and for z0 such that the following

conditionissatisfiedComputen0andz0fortheconditionsoftheproblem

Considerthefactthatthequantity

isapproximatelynormallydistributedwithparameters

wherek=nndash1ComparetheresultwiththatofProblem455457 Using the binomial and Poisson distributions construct the plan of

doublecontrolforn1=n2=30v1=3v2=5v3=8ifalotisconsideredgoodwhentheproportionofdefectiveitemsispleP0=010anddefectivewhenpgep1=020ForthevaluesαandβfoundforthebinomialdistributionconstructtheplansofsingleandsequentialcontrolcompareallthreemethodsaccordingtotheaveragenumberoftestsForthesequentialcontrolfindnminforagoodlot

andadefectivelotandcomputetheexpectationofthenumberoftestsM[n|p]458 Construct the control plans by the methods of single and sequential

samplingforlargelotsofradiotubesifalotwithproportionofdefectiveitemsplep0=002isconsideredgoodandwithpgep1=007 isconsidereddefectiveTheproducerrsquosriskisα=00001and theconsumerrsquos risk isβ=001For theplanofsequentialcontroldeterminenminforagoodlotandadefectiveonefindtheaveragenumberoftestsM[n|p]andtheprobabilitiesP(nleM[n|p0)P(nle2M[n|p0])

459 The time of operation T (in hours) of a transformer obeys anexponential distributionwith an intensity of failuresλAssuming that λt0 lt 1constructtheplansofcontrolbysinglesamplingandsequentialanalysisforα=010β=010Forthesinglecontrolfindtheacceptancenumbervandthesizen0ofthesampleifthetestingperiodofeachtransformerist0=500100020005000hours(ReplacethePoissondistributionbyachi-squaredistribution)Forthe sequential control take a fixed sample sizen0 corresponding to t0 = 1000hours and find the average testing timeof each transformerM[T |λ] Assumethatalotoftransformersisgoodiftheintensityoffailuresλleλ0=10ndash5hoursndash1

anddefectiveifλgeλ1=210ndash5hoursndash14510Alargelotofelectricalresistorsissubjectedtocontrolforα=0005β

=008thelotisconsideredgoodiftheproportionofdefectiveresistorsisplep0=002anddefectiveifpgep1=010Applyingachi-squaredistributioninsteadofaPoissononefindthesizen0andtheacceptancenumbervforthemethodofsingle sampling construct the plan of sequential control for a good lot and adefective lot compute the expectation of the number of tested items and theprobabilitiesP(nltn0)P(nlt(12)n0)

4511BeforeplantinglotsofseedpotatoesarecheckedforrottingcentersAlot of seed potatoes is considered good for planting if in each group of 10potatoesthereisatmostonespotandbadiftherearefivespotsormore

AssumingthatthenumberofspotsobeysaPoissondistributioncomputeaandαandβforthemethodofdoublesamplingifn1=40n2=20v1=4v2=12v3=14Fortheresultingvaluesofαandβconstructtheplansofsingleandsequential controlCompare the efficiencies of all threemethods according tothemeanexpendituresofseedpotatoesnecessarytotest100lots

4512Thequalitycharacteristicinalotofelectricalresistorswhoserandomvaluesobeyanormaldistribution lawwithaknownmeanof200ohms is thestandarddeviationσandthelotisacceptedifσleσ0=10ohmsanddefectiveif

σgeσ1=20ohmsConstructthecontrolplansbythemethodofsinglesamplingwithn0=16v=1292anddoublesamplingwithn1=n2=13v1=v3=12v2=infinFortheresultingvaluesofαandβ(inthecaseofsinglecontrol)constructtheplan of sequential control Compare the efficiencies of all three methods ofcontrolaccordingtotheaveragenumberof testsComputenminforthepoorestamongthegoodlotsandthebestamongthedefectivelots

4513SeverallotsofnylonaretestedforstrengthThestrengthcharacteristicX measured in gdenier (specific strength of the fiber) obeys a normaldistributionwithstandarddeviationσ=08gdenierAlotisconsideredgoodifXgex0=54gdenierandbad ifXgex1=49gdenierConstruct theplanofstrengthcontrolbysinglesamplingwithn0=100andv=51Fortheresultingvalues of α and β construct the plan of control by the method of sequentialanalysiscomputethemeanexpenditureinfibersandtheprobabilitiesP(nltn0)P(nlt(12)n0)

4514Itisknownthatiftheintensityoffailuresisλleλ0=001thenalotofgyroscopes is considered reliable if λ ge λ1 = 002 the lot is unreliable andshould be rejected Assuming that the time T of reliable operation obeys anexponentialdistributionandtakingα=β=0001constructtheplansforsingle(n0v)andsequentialcontrolsaccordingtotheleveloftheparameterλFindtheaveragenumberoftestedgyroscopesM[n|λ]forthecaseofsequentialcontrol

4515AlargelotofcondensersisbeingtestedThelotisconsideredgoodiftheproportionofunreliablecondensersisplep0=001forpgep1=006thelotis rejected Construct the plan of single control (n0 v) for the proportion ofunreliableitemssothatα=005β=005

Toestablishthereliabilityeachtestedcondenserbelongingtotheconsideredsampleissubjectedtoamultiplesequentialcontrolforαprime=00001βprime=00001and a condenser is considered reliable if the intensity of failures λ le λ0 =00000012andunreliable forλgeλ1=00000020hoursndash1 (n is thenumberoftests used to establish the reliability of a condenser for givenαprime and βprime) Oneassumesthatthetimeofreliableoperationofacondenserobeysanexponentialdistribution

4516 Construct the plans of single and sequential controls of complexelectronicdeviceswhosereliabilityisevaluatedaccordingtotheaveragetimeofunfailing (reliable)operation If geT0=100hours a device is consideredreliableand ifTleT1=50hoursunreliable It isnecessary thatα=β=010

ConsiderthatforafixedtestingtimetTadeviceisacceptediftTm= gevandrejected if lt v wherem is the number of failures for time t and v is theacceptancenumber in the caseof single control (n0=1 in caseof failure thedevice is repaired and the test is continued) In this case tT obeysapproximately a Poisson distribution In the case of sequential control thequantitytdependsontheprogressofthetest

(a)Determine the testing time tT and the acceptancenumberv for a singlecontrol

(b)FortheplanofsequentialcontrolreducetheconditionforcontinuationofthetestsInBltInγ(tm)ltInAtotheformt1+mt3gttgtt2+mt3Fort1t2t3obtainpreliminarygeneralformulas

(c)Inthecaseofsequentialcontroldeterminetheminimaltestingtimetminforthepoorestofthegoodlotsandthebestoftherejectedones

46 DETERMINATIONOFPROBABILITYCHARACTERISTICSOFRANDOMFUNCTIONSFROMEXPERIMENTALDATA

BasicFormulasThe methods of determination of the expectation the correlation function

andthedistributionlawsoftheordinatesofarandomfunctionbyprocessingaseriesofsamplefunctionsdoesnotdifferfromthemethodsofdeterminationofthecorrespondingprobabilitycharacteristicsofasystemofrandomvariablesInprocessing the sample functions of stationary random functions instead ofaveraging the sample functions one may sometimes average with respect totime ie find the probability characteristics with respect to one or severalsufficientlylongrealizations(theconditionunderwhichthisispossibleiscalledergodicity) In this case the estimates (approximate values) of the expectationand correlation function are determined by the formulas

whereTisthetotaltimeofrecordingofthesamplefunctionSometimesinstead

ofthelastformulaoneusesthepracticallyequivalentformula

Inthecasewhentheexpectation isknownexactly

If and aredeterminedfromtheordinatesofasamplefunctionofarandom function at discrete time instants tj = (j ndash 1) Δ the corresponding

formulasbecome

or

whereτ=lΔT=mΔFornormalrandomfunctionsthevariances and maybeexpressed

in terms ofKx(τ) In practical computations the unknown correlation function

Kx(τ)intheformulasforD[ ]andD[ ]isreplacedbythequantity

When one determines the value of the correlation function by processingseveral sample functions of different durations one should take as the

approximatevalueoftheordinatesof thesumofordinatesobtainedbyprocessing individual realizationswhoseweights are inversely proportional tothevariancesoftheseordinates


Example461Theordinatesofastationaryrandomfunctionaredeterminedby photographing the scale of the measuring instrument during equal timeintervalsΔDeterminethemaximaladmittedvalueofΔforwhichtheincreasein

thevarianceof comparedwith thevarianceobtainedbyprocessing thecontinuousgraphofrealizationofarandomfunctionwillbeatmostδpercentiftheapproximatevalueof =aendashα|τ|andthetotalrecordingtimeTis 1αItisknownthat =0andthefunctionX(t)canbeconsiderednormal

SOLUTION Since = 0 by use of the continuous recording the value of

isdeterminedbytheformula

Forfindingthevarianceof wehave

If after integration we eliminate the quantities containing the small (byassumption)factorendashαTweget

Iftheordinatesoftherandomfunctionarediscretethevalueof is

DeterminingthevarianceofK2(0)wefindthat

whereforthecalculationoftheexpectationoneusesapropertyofmomentsofsystemsofnormalrandomvariables

Usingthevalueof weobtain

ThelimitingvalueofΔisfoundfromtheequation

thatisfromtheequation

ForαΔ 1weobtainapproximately

PROBLEMS

461Provethatthecondition

isnecessaryinorderthatthefunctionX(t)beergodic462Verifywhethertheexpression

maybetakenasanestimateofthespectraldensityifX(t)isanormalstationary

randomfunction( =0)and 463 To determine the estimate of the correlation function of a stationary

normal stochastic process X(t) ( = 0) a correlator is used that operates

accordingtotheformula

DerivetheformulaforD[ ]464 Determine the expectations and the variances of the estimates of

correlation functions defined by one of the formulas

where ifX(t)isanormalrandomfunction465ThecorrelationfunctionofthestationarystochasticprocessX(t)hasthe

form

Findthevariancefortheestimateoftheexpectationdefinedbytheformula

466 The spectral density is found by a Fourier inversion of the

approximate value of the correlation function Determine D[ ] as a

functionofωiftheprocessisnormalandtosolvetheproblemonemayuse

insteadofKx(Δ)inthefinalformula467ThecorrelationfunctionKx(Δ)determinedfromanexperiment isused

for finding the variance of the stationary solution of the differential equation

Determinehowσywillchangeifinsteadoftheexpression

representingasufficientlyexactapproximationofKx(Δ)oneuses

where α1 and β1 are chosen such that the position of the first zero and the

ordinate of the first minimum of the expression of coincide with thecorrespondingquantitiesforKx(τ)

468AnapproximatevalueofKx(τ)isusedtofindD[Y(t)]where


whichapproximatesquiteaccuratelytheexpressionKx(τ)oneuses

whereαandβarechosensuchthatthepositionofthefirstzerosandthevalueofthefirstminimumofthefunctions and coincide

469Thecorrelationfunctionfortheheelangleofashipcanberepresentedapproximatelyintheform

wherea=36deg2α=005secndash1andβ=075secndash1

DetermineD[ ] for τ = 0 and τ = 3 sec ifΘ(t) is a normal random

function and is obtained by processing the recorded rolling of the shipduringtimeT=20minutes

4610Theordinateoftheestimateofthecorrelationfunctionforτ=0is100cm2andforτ=τ1=419secItsmodulusattainsamaximumcorrespondingtoa negative value of 415 cm2 According to these data select the analytic

expression for (a) in the form =

(b)intheform Determine the difference in the values of the first zeros of the functions

inthesetwocases4611 Determine D[ ] for τ= 0 209 418 and 1672 sec if

wherea=25deg2α=012secndash1β=075secndash1andΘ(t)isanormalrandomfunction =0Todetermine oneusesa10mrecordingofΘ(t)where1cmofthegraphalongthetimeaxiscorrespondsto1sec

4612ThegraphofasamplefunctionoftherandomfunctionX(t)isrecordedon a paper tape by using a conducting compound passing at constant speedbetweentwocontactsoneshiftedwithrespecttotheotherbyτsecondsalongthetimeaxisThecontactsareconnectedtoarelaysystemsothattherelayturnsonastopwatchwhentheordinatesofthesamplefunctionatthepointswherethecontactsarelocatedhavethesamesignandturnsitoffotherwiseShowthatif= 0 and X(t) is a normal stationary random function the estimate of itsnormalized correlation function can be determined by the formula

where t1 is the total reading of the stopwatch and t is the total time the tapemoves

4613UndertheassumptionsoftheprecedingproblemdetermineD[ ]

if for the determination of one uses the graph of the sample functioncorresponding to the recording time T = 10 minutes

4614AsaresultofprocessingthreesamplefunctionsofasinglestationaryrandomfunctionX(t)fordurationsT1T2andT3threegraphsofestimatesofthecorrelationfunctionwereobtainedAssumingthattheprocessisnormalderivetheformulaforfindingtheordinatesoftheestimateofthecorrelationfunction

Usealltheexperimentaldatawiththeconditionthatthevarianceoftheerror is minimal if for each sample function the estimate of the correlationfunction is given by the formula

4615Determine variance of the estimate for the correlation function of a

normalstochasticprocesswithzeroexpectationiftofind onetakestheordinates of the sample function of the random function during equal timeintervalsΔ thedurationof recording isT=mΔand in the final formulaKx(τ)

maybereplacedby 4616Theordinatesofarandomfunctionaredeterminedbyphotographing

thescaleofaninstrumentduringequaltimeintervalsΔ=1secDeterminethe

ratioofD[ ]tothevarianceobtainedbyprocessingthecontinuousgraphofthesamplefunctionif(τisexpressedinseconds)theprocessisnormalandtheobservationtimeT=5minutes

4617AnapproximatedeterminationoftheordinatesofasamplefunctionofastationaryrandomfunctionX(t)withzeroexpectationandaknowncorrelationfunction Kx(τ) is given by the formula

whereAjBjaremutuallyindependentrandomvariableswithunitvariancesandzeroexpectationsandTisaknownnumberDeterminetheconstantsαjsothat

where is the correlation function corresponding to the precedingapproximate expression for X(t) Determine the magnitude of ε for optimalvaluesoftheconstants

4618Todecrease the influence of the randomvibration of the frameof amirror-galvanometerusedtomeasureaweakcurrentthereadingsarerecordedduringT=10secandthevaluejoftheaveragerecordedordinateisconsideredtobetherequiredintensityofthecurrentFindthemeanerroroftheresultifthevibrationoftheframeisdescribedbythecorrelationfunctionoftheintensityofcurrentJ(t)

where

1 If the variable considered is normal then the unbiased estimate for the standard deviation isdeterminedfromtheformula

2ReferencesforthetablenumbersfollowedbyTarefoundonpages471ndash473

ANSWERSANDSOLUTIONS

I RANDOMEVENTS


11BydefinitionAcupA=AAA=A12TheeventAisaparticularcaseofB13B=A6C=A514(a)AcertaineventU(b)animpossibleeventV15(a)Atleastonebookistaken(b)atleastonevolumefromeachofthe

threecompleteworksistaken(c)onebookfromthefirstworkorthreebooksfrom the second or one from the first and three from the second (d) twovolumesfromthefirstandsecondworksaretaken(e)atleastonevolumefromthethirdworkandonevolumefromthefirstworkandthreefromthesecondoronefromthesecondandthreefromthefirst

16Theselectednumberendswith517 meansthatall itemsaregood means thatoneornoneof themis

defective18Usingthepropertiesofevents(BcupB=BBB=BBcup =UBU=B

B =VBcupV=B)wegetA=BC19(a)AmeansreachingtheinterioroftheregionSA meanshitting the

exteriorofSAThenAcupB=UthatisA=VB=U(b)ABmeansreachingtheregionSABcommontoSAandSB meansfallingoutsideSAThenAB=VthatisA=UB=V(c)ABmeansreachingthecommonregionSABAcupBmeanshittingSAcupBSAB=SAcupBonlyifSA=SBthatisA=B

110X=111Usetheequalities = Bcup =A cup 112TheequivalenceisshownbypassingtothecomplementaryeventsThe

equalitiesareprovedbypassagefromnton+1

113Nosince 114Usetheequality115Cmeansatie116117

118C=(A1cupA2)(B1B2cupB1B3cupB2B3)

2 ADIRECTMETHODFOREVALUATINGPROBABILITIES

21p=rmn224923p=025sincethefirstcardmaybelongtoanysuit24165asymp000013252324026 The succession of draws under such conditions is immaterial and

thereforep=2927Onemayconsiderthatforcontroltheitemsaretakenfromthetotallotp

=(nndashk)(n+mndashk)28Onemayconsiderone-digitnumbers(a)02(b)04(c)00429(a)N=a+10bThisconditionissatisfiedonlyifaisevenanda+bis

divisibleby9p=118(b)N=a+10b+100cThisnumbershouldbedivisibleby4andby9thatisa+b+cisdivisibleby9a+2bisdivisibleby4(m=22)p=11360

210

211

21221303

214

215

216pk= (k=12345)Pl=00556p2=00025p3=085middot10ndash4p4=02middot10ndash5p5=02middot10ndash7

217

218

219220 Thefavorablecombinations(a)(777)(b)(99

3)(966)(c)(2811)(2910)(3711)(3810)(4611)(4710)(489)(678)andthereforem=4+2middot4middot +43middot8=564p=0079

221

222 It is necessary to get n ndashm nickels from 2n buyers The number ofpossible cases is whereN is the number ofcases when it is impossible to sell 2n tickets

is the number of cases in which thefirstnickelcamefromthe(2m+2)ndbuyer isthenumberofcasesinwhichthefirstnickelcamenotlaterthanfromthe(2m+1)stbuyerandthesecondnickelfromthe(2m+4)thbuyerandsoon


31

32

33 34ConstructionAB isasegmentof length2hC is thecenterof thedisk

AD andBE are tangents to the disk located on one side of the lineAC ThetrianglesADCandBECcoincidebyrotationwithangleφ=angDCEthereforeangACB=φh=ltan(φ2)p=(1π)arctan(hl)

35

36(a)00185(b) 37(a)016(b)0638xisthedistancefromtheshoretotheboatandy(withthecorresponding

sign)fromtheboattothecourseoftheshipPossiblevaluesxle1middotυforylt0x+yle1 middotυ forylt0 |y |lex (υ is the speedof theboat1=1hour)Thefavorablevalues|y|le(l3)υp=59

39k(2ndashk)310x=ALy=AMPossiblevalues0lex+ylelThefavorablevalues|y

ndashx|lexp=075311TwosegmentsxyPossiblevalues0lex+ylelFavorablevaluesxle

12ylel2x+ygel2p=14312TwoarcsxyPossiblevalues0le(x+y)le2πRFavorablevaluesxle

πRyleπRx+ygeπRp=14313SegmentsxyzPossiblevalues0le(xyz)lelFavorablevaluesx+

ygezx+zgeyy+zgexp=12314AM=xMN=yPossiblevalues0lex+ylelFavorablevaluesxlea

yleax+ygelndashaForl3lealel2p=[1ndash(3al)]2forl2lealelp=1ndash3[1ndash(al)]2

315xisanarbitraryinstant0lexle12minutesTheinstantsofarrivalofabusbelongingtolineAx=048theinstantsofarrivalofabusoflineByy+6where0leyle4(a)Favorablevaluesfor0ltyle2wehaveyltxle46+ylexle12forygt2wehaveyltxlt8ory+6ltxlt12p=23(b)Favorablevalues2lexle46lexle810lexle124+ylexle6+yforylt2wehave0ltxleyandforygt2yndash2lexleyp=23

316xyarethetimesofarrivaloftheshipsPossiblevalues0lexle240leyle24Favorablevaluesyndashxle1xndashyle2p=0121

317 318xisthedistancefromtheshoretothefirstshipandythedistancetothe

second ship Possible values 0 le (x y) leL The favorable region |x ndash y | le isobtainedbypassagetotherelativemotion(thefirstship

remains fixed and the second ship moves with speed υ = υ2 ndash υ1) for

for

319 (a) p = 1 ndash (1920)2 = 00975 (b) x y z are the coordinates of theinflectionpointsPossiblevalues0le(xyz)le200Favorablevalues|xndashy|le10|xndashz|le10|yndashz|le10p=1ndash(180200)3=0271

320321

322xisthedistancefromthemidpointoftheneedletothenearestlineandφistheanglemadebythelinewiththeneedlePossiblevalues0lexleL20leφleπFavorablevaluesxle(l2)sinφp=2lLπ

323Possiblevalues|a|len|b|lem(a)Favorablevaluesblea2Formgen2

Formlen2

Therootswillbepositiveifale0bge0Formgen2p=n212mformlen2p=14ndash 6n(b)Therootsoftheequationwillberealifb2+a3le0Theregionforfavorablevaluesofthecoefficientsale0b2lendasha3

Forn3lem2

Forn3gem2

324LetAandBbethepositionsofthemovingpointandthecenterofthecircleuandvtheirvelocityvectorsandrthedistanceABFromthepointBweconstructacircleofradiusRWeconsiderthatβgt0ifthevectorvliestotheleftofthelineABndashπleβleπFromthepointAweconstructtangentstothecircleofradiusR The pointA reaches the interior of the circle if the relative velocityvectorfallsintotheresultingsectorwhoseangleis2εε=arcsin(Rr)FromAweconstructthevectorndashvLetObetheendpointofthisvectorFromOwedrawacirclewhose radius coincides inmagnitudewith thevelocityof thepointAThepointAwilllieinthecircleonlyifthevectorundashvliesinthesectorLetugtvThentherequiredprobabilitywillbe(Figure41)p=α2πTodetermineαweset Thenα=2ε+δndashγUsingtheequalities

weobtain

ThepresentformulaisvalidforanyβForυgtutheproblemmaybesolvedsimilarlybutinthiscaseoneshouldconsiderseveralcases(1)|β|geε+(π2)p=0(2)(π2)+εle|β|geε(a)foruleυsin(|β|ndashε)weshallhavep=0(b)forυsin(|β|ndashε)leuυsin(|β|+ε)wehave

FIGURE41

(c)forugtυsin(|β|+ε)weshallhave

(3)|β|leε(a)foruleυsin(εndash|β|)weshallhavep=1(b)for

weshallhave

(c)forugtυsin(ε+|β|)weshallhave


41p=1ndash03middot02=094

4243p=(1ndash02)3=051244025145p=1ndash(1ndash03)(1ndash022)=032846p(1ndashp)nndash1471ndash05nge09nge4481ndash(1ndashp)4=05pasymp0159

49

410411FromtheincompatibilityoftheeventsitfollowsthatP(A |B)=0and

P(B|A)=0thatistheeventsaredependent412P1P2413p=07middot0912=0197414p=072(1ndash062)=0314415075416p1=09middot08middot07middot09asymp045p2=072middot08asymp039417(a)01=(p1p3)nthatisn=ndash1(logp1logp3)(b)p=1ndash(1ndash(1ndashp1p3)3(1ndashp2p4)3

418ItfollowsfromtheequalityP(A)P(B|A)=P(B)P(A|B)

419

420421

422

423

424

425P(A)=P(B)=P(C)=

thatistheeventsarepairwiseindependent

thatistheeventsarenotindependentintheset426No(seeforexampleProblem425)427p=nnn

428

429

430

431

432433Leta1a2hellipanbethebuyerswhohavefive-dollarbillsandb1b2hellip

bmthosewithten-dollarbillsandsupposethattheirnumberscoincidewiththeirorderinthelineTheeventAkmeansthatonewillhavetowaitforchangeonlybecauseofbuyerbk(k=12hellipm)

434ItmaybesolvedasonesolvesProblem433

435 The first ballot drawn should be cast for the first candidate Theprobabilityofthisisn(n+m)ThentheballotsmustfollowinsuccessionsothatthenumberofdrawnvotescastforthefirstcandidateisalwaysnotsmallerthanforthesecondoneTheprobabilityofthiseventis(nndashm)n(seeProblem433)


510035205553pk= pkj542(rR)255112656

57P(A )=P(A)ndashP(AB)58P(B)=P(AB)+P( B)=[P(A)+P( )]P(B|A)=P(B|A)59P(B)=P(A)+P(B )geP(A)510032351105512npqmndash1513(a)13(b)56514AmeansthatthefirsttickethasequalsumsBthesecondticket

(a)P(AcupB)=2P(A)=01105(b)P(AcupB)=2P(A)ndashP2(A)=01075515FromP(AcupB)le1itfollowsthatP(B)ndashP(AB)leP( )or

516FromZ=XcupYitfollowsthatZleX+|Y|ZgeXndash|Y|P(Zle11)geP(Xle10and|Y|le1)=P(Xle10)+P(|Y|le1)ndashP(Xle10or|Y|le1)ge09+095ndash1=085P(Zge9)ge005P(Zle9)le095

517044and035518p(2ndashp)519pB=01+09middot08middot03=0316pc=09(02+08middot07middot04)=03816

520521pBasymp08pcasymp02

522G(m+n)=G(m)+[1ndashG(m)]G(n|m)

523

Anothersolutionp1+p2=1p2=(12)p1thatisp1=23p2=13524 P1 + p2 + p3 = 1

525p+q=1q= pp=

526527 p1 is the probability of hitting for the first marksman p2 is the

probabilityofhittingforthesecondmarksmanp1+p2=102p2=08middot03p1p=p1=0455

528UsetheconditionofProblem112529Ifwecalculatethenumberofidenticaltermsweget

530Usingtheequality fromProblem112andthegeneralformulafortheprobabilityofasumofeventsweobtain

However according to Problem 112 we have andhence for any s Also considering theequality

wegettheformulaindicatedintheassumptionoftheproblem531Usetheequality

andtheformulafromtheconditionofProblem530

532533 The probability that m persons out of n will occupy their seats is

The probability that the remaining n ndashm personswillnotsitintheirseatsis

534TheeventAjmeansthatnopassengerwillenterthejthcar

andsoonUsingtheformulafromtheanswertoProblem529weobtain

535Thefirstplayerwinsinthefollowingncases(1)inmgameshelosesnogame(2)inmgameshelosesonebutwinsthe(m+1)stgame(3)inm+1gameshelosestwobutwinsthe(m+2)ndgamehellip(n)inm+nndash2gameshelosesnndash1andthenhewinsthe(m+nndash1)stgame

536Thestackisdividedintheratiop1p2ofprobabilitiesofwinningforthefirstandsecondplayers

537TheeventAmeansthatthefirsttoldthetruthBmeansthatthefourthtoldthetruth

Let pk be the probability that (in view of double distortions) the kth liartransmittedthecorrect informationp1=13p2=59p3=1327p4=4181P(A)=p1P(B|A)=p3P(B)=p4p=1341

538WereplacetheconvexcontourbyapolygonwithnsidesTheeventAmeans that line Aij will be crossed by the ith and jth sides

wherebeingthe

probability that theparallel linesarecrossedbythekthsideof length lkFromthesolutionofBuffonrsquosProblem322 it follows that =2lkLπpprime= (1 Lπ)

lk Since this probability is independent of the number and size of thesideswehavep=sLπ


61

6263H1meansthatamongtheballsdrawntherearenowhiteballsH2means

thatoneballiswhiteandH3thatbotharewhite

64Hj1meansthatawhiteballisdrawnfromthejthurn

Consider

ThenP(Hj+11)=m(m+k)Thereforep=m(m+k)6507662967022568075690332610TheeventAmeansgettingacontactThehypothesisHkmeans thata

contactispossibleonthekthband(k=12)Letxbethepositionofthecenteroftheholeandythepointofapplicationofthecontact

Thecontactispossibleonthefirstbandiffor25lexle35|xndashy|le5for15lexle2520leylex + 5 for 35lex le 45x ndash 5ley le 45ThusP(A |H1)=115SimilarlyP(A|H2)=114p=0045

611TheeventAmeans that s calls come during the time interval 2tThehypothesisHk(k=01hellips)meansthatduringthefirstintervalkcallscameP(Hk)=Pt(k)Theprobability thatsndashk calls comeduring the second intervalwillbe

612ThehypothesisHkmeansthattherearekdefectivebulbsP(Hk)=16(k=01hellip5)TheeventAmeansthatall100bulbsaregood

613ThehypothesisHkmeansthattherearekwhiteballsintheurn(k=01hellipn)theeventAmeansthatawhiteballwillbedrawnfromtheurn

614ThehypothesisHk(k=0123)meansthatknewballsaretakenforthefirstgameTheeventAmeansthatthreenewballsaretakenforthesecondgame

615

616617P(A)=P(AB)+P(A )=P(B)P(A|B)+P( )P(A| )

Theequalityisvalidonlyinseveralparticularcases(a)A=V(b)B=U(c)B=A(d)B= (e)B=VwhereUdenotesacertaineventandVan impossibleone

618BytheformulafromExample62itfollowsthatmasymp13pasymp067619Inthefirstregionthereareeighthelicopterspasymp074


71

7273ThehypothesisH1meansthattheitemisastandardoneandH2thatitis

nonstandardTheeventAmeansthattheitemisfoundtobegood

74 The hypothesesHk (k = 0 1hellip 5) means that there are k defective

itemsTheeventAmeansthatonedefectiveitemisdrawn

ThemostprobablehypothesisisH5thatistherearefivedefectiveitems

75P(H0|A)= =0214(seeProblem612)76TheeventA denotes thewinofplayerD thehypothesisHk (k=12)

meansthattheopponentwasplayerBorC

77Thesecondgroup78TheeventAmeansthattwomarksmenscoreahitHkmeansthatthekth

marksmanfails

79TheeventAmeansthattheboariskilledbythesecondbullet

ThehypothesisHkmeansthatthekthmarksmanhit(k=123)

710Thefourthpart711p=nk(1+2k+middotmiddotmiddot+nk)712TheeventsareM1thatthefirsttwinisaboyM2thatthesecondisalso

aboyThehypothesesareH1thatbothareboysH2thatthereareaboyandagirl

713AkmeansthatthekthchildbornisaboyandBkthatitisagirl(k=12)P(A1A2)+P(B1B2)+2P(A1B2)=1P(A1A2+B1B2)=4P(A1B2)Therefore

714511715Oneoccurrence716HypothesisH1meansthatthefirststudentisajuniorandH2meansthat

heisasophomoreAdenotestheeventthatthesecondstudenthasbeenstudyingformoretimethanthefirstBmeansthatthesecondstudentisinthethirdyear

71714and211718ThehypothesesHk(k=01hellip8)meanthateightoutofkitemsare

nondefective A denotes the event that three out of four selected items arenondefective


81(a)094=0656(b)094+4middot01middot093=0948

82 (a) (b)

83

840178506486(a)0163(b)035387p=1ndash(084+4middot083middot02+5middot082middot022+2middot08middot023)072middot06=0718

8889p=1ndash(074+4middot073middot03middot04)=0595810HypothesisH1means the probability of hitting in one shot is 12H2

means that this probability is 23 The eventA means that 116 hits occurredP(H1|A)asymp2P(H2|A)thatisthefirsthypothesisismoreprobable

811SeeTable113TABLE113

81202813073814Rn1asymp1ndashendash002n(ngt10)SeeTable114

TABLE114

815p=1ndash09510=04816p=1ndash095=041817

818(a) (b)02438190488820AdenotestheeventthattwogooditemsareproducedThehypothesis

Hkmeansthatthekthworkerproducestheitems(k=123)

821(a) (b)3p4ndash4p3+ =0p=0614822

823 8240784825The200wones(R61=0394R102=0117)82606482702816828

829 830Werequire

831 We require

832P4 0=03024P4 1=04404P4 2=02144P4 3 = 00404P4 4 =00024

833026834015983595144

836n=29837nge10838nge1683988408841μ=4p=0251842μ+=3μndash=1p=3281


91p=P5221+2P5320=5024392p=P3111+P3210+P3120=0245

93(a) (b)

94 95

96 (a) (b)

97(a)

98p=pnpk=pkndash1middot +(1ndashpkndash1) =05p=0599 let pk be the probability of a tie when 2k resulting games have been

playedpk+1=(l2)pk(k=01hellip)p0=1pnndash1=(12)nndash1p=(l2)pnndash1=l2n

910ThenumbernshouldbeoddLetpkbetheprobabilitythatafter2k+1gamestheplayisnotterminatedp0=1

911LetpkbetheprobabilityofruinofthefirstplayerwhenhehaskdollarsAccordingtotheformulaoftotalprobabilitypk=ppk+1+qpkndash1Moreoverp+q=1p0=1pn+m=0Consequentlyq(pkndashpkndash1)=p(pk+1ndashpk(1)p=qThenpk=1ndashkcc=1(n+m)thatispI=m(n+m)pII=n(n+m)(2)pneqThenpkndashpkndash1=(pq)k(p1ndash1)Summingtheseequalitiesfrom1tonandfrom1ton+mweobtain

Thus

912P=PmPm=0formgenPn=12nndash1Pm=12nfornltmlt2nndash1InthegeneralcasePmisdeterminedfromtherecurrentformula

whichisobtainedbytheformulaoftotalprobabilityInthiscasethehypothesisHkmeansthatthefirstopponentofthewinnerwinskgames

913PkistheprobabilitythatexactlykgamesarenecessaryFork=1234 5 Pk = 0 P6 = 2p6 = 125

P9 = 725 P10 = 6329 (a)(b)ifnisoddthenPn=0ForevennPn=(1

2)p(n ndash 1)2wherepk is theprobability thatafter2kgames theopponentshaveequal numbers of points thatis

914Expand(1ndashu)ndash1intoaseriesandfindthecoefficientofum915ThesameasinProblem914916 The required probability is the constant term in the expansion of

generatingfunction

917 The required probability is the sum of the coefficients of u raised topowersnotlessthanmintheexpansionofthefunction

Forn=m=3p=0073918Therequiredprobabilityistwicethesumofthecoefficientsofu4inthe

expansionofthefunction

919 (a) The required probability pchamp is the sum of the coefficients ofnonnegativepowersofuintheexpansionofthefunction

(b)theprobabilityofthecomplementaryeventisthesumofthecoefficientsofuwhosepowersrangefromndash4to3intheexpansionofthefunction

920(a)TherequiredprobabilityPmisfoundwiththeaidofthegeneratingfunction

Using the equality weobtain

andtheseriesiscutoffifmndash6kltn(b) Usingtheequalityweobtain

Forn=10m=20

921Thedesiredprobabilityisthecoefficientofu21intheexpansionofthefunction

922(a)pNisthecoefficientofuNintheexpansionofthefunction

andtheseriesiscutoffwhenNndashmsltn

(comparewithProblem920)923

924HypothesisHkmeansthatthenumbersofheadsforthetwocoinsfirstbecomeequalafterktossesofbothcoins(k=12hellipn)theeventAmeansthatafter n throws the numbers of heads become equal (previous equality is notexcluded)

Consequently Using successful values fornonecanfindp=P(Hn)Letwherepnndashj=P(A|Hj)Addingtogetherthetermscontainingunweobtain

925 Let μ be the number of votes cast for a certain candidate Theprobabilityofthisis Theprobabilitythatatmostμvotesarecast for this candidate is The probability that among kcandidates lndash1 receiveat leastμvoteskndash l ndash 1 personsget nomore thanμvotesandtworeceiveμvoteseachis

926Theprobabilityofwinningonepointfortheservingteamis23

or

ThenumbersPkandQkaregiveninTable115

TABLE115

(b)(c)letαkbe theprobabilityofscoring14+kpointsoutof28+2k for the

firstteam(serving)whichwinsthelastballβkbeingtheanalogousprobabilityforthesecondteam

thatis

II RANDOMVARIABLES





104(a)P(X=m)=qmndash1p=l2m(b)oneexperiment105X1istherandomnumberofthrowsforthebasketballplayerwhostarts

thethrowsandX2isthesameforthesecondplayer


107P(X=m)=qmndash4p=l2mndash3 forallmge4since theminimal randomnumber of inclusions is four and occurs if the first device included ceases tooperate

108109 1010P(X=m)=1ndash2middot025mforallmge11011P(X=k)=(1ndashpω)kndash1pωforallkge11012P(X=m)=(np)mmendashnpforallmge01013SeeTable120

TABLE120



111

112113

114

115(a)σ(b) (c)

116 (a) (b) xp = ndash x0 ln (1 ndash

p)1m(c)

117 (a) 10 (b) where

118 (a) (b) (c)

119

1110(a)F(x)= + arctanx(b)P(|x|lt1)=

1111p=

1112p= 1113 introduce the random variable X denoting the time interval during

whichatubeceasestooperateWritethedifferentialequationforF(x)=P(Xltx) the distribution function of the random variable X The solution of thisequationforx=lhastheformF(l)=1ndashendashkl]

1114

1115


121 ndashp122 a=18 b=17 B=20theminimalnumberofweighingswillbe

inthecaseofsystem(b)123M[X]=2D[X]=11124Toprovethis it isnecessarytocomputeM[X]=dG(u)du|u = 1where

G(u)=(q1+p1u)(q2+p2u)(q3+p3u)125WeformthegeneratingfunctionG(u)=(q+pu)nM[X]=Gprime(1)=np

126 127Forthefirst711forthesecondndash711coinsthatisthegameislost

bythesecondplayer128Considerabandcas theexpectedwinsofplayersABandCunder

theassumptionthatAwinsfromBForthesequantitiesthereobtaina=(m2)+(b2)c=a2b=c2formingasystemofequationsfortheunknownsabandcSolvingthesystemweobtaina=(47)mb=(l7)mc=(27)mInthesecondcase we obtain for the players A B and C (514)m (514)m (27)mrespectively

129

1210

1211

1212M[X]=kpD[X]=[k(1ndashp)p]Theseries

issummedwiththeaidoftheformula

whereq=1ndashp1213 (a)M[m] = ω where ω = 1(1 ndash endash α) (b)M[m] = ω + 1 For

summationoftheseriesweusetheformulas

1214M[X]=l[p1+p2p3(1ndashp1)]=455wherep1=018p3=p2=0221215M[X]=4(23)12161217Findthemaximumofthevarianceasafunctionoftheprobabilityof

occurrenceofanevent1218μ3=np(1ndashp)(1ndash2p)vanishesatp=0p=05andp=11219 Treat the variance as a function of probability of occurrence of an

event1220Inbothcasestheexpectednumberofblackballsinthesecondurnis5

andofwhiteballsis4+1210inthefirstcaseand4+endash5inthesecondcase1221Twodollars1222Forplt341223M[X]=[(n2ndash1)3n]aForfindingtheprobabilitiespk=P(X=ka)that

therandomlengthoftransitionequalskausetheformulaoftotalprobabilitiesandtakeashypothesisAithefactthattheworkerisattheithmachine

1224q=09P10=1ndashq10asymp0651

1225M[X]=3212261227y=12py=65dollars1228M[X]=nmD[X]=n(m+n)m2

1229 limkrarrinfinXk=[(M+M1)(N+N1)]NWritetheequationoffinitedifferencesforthe expected number of white balls Xk contained in the first urn after kexperiments

1230

1231 =qpD[X]=q2p2+qpwhereq=1ndashp1232

since


131M[X]=aD[X]= E=

132M[X]=0D[X]=

133

134 135

136 137M[X]=D[X]=m+1

138 139M[X]=0D[X]=2

1310 M[X]=(α+1)βD[X]=β2(α+1)1311

1312

To calculate the integral use the change of

variables leading to theB-functionandexpress the latterintermsoftheT-function

1313

1314Usetherelation

1315M[T]=1γNoticethatp(t)isthedistributionfunctionoftherandom

timeofsearch(T)necessarytosighttheship1316m(t)=m0endashpiConsider the fact that theprobabilityofdecayofany

fixed atom during the time interval (t t + Δt) is p Δ t and work out thedifferentialequationform(t)

1317TII=(lp)(log2)(loge)UsethesolutionofProblem13161318[P(Tlt )][P(Tgt )]=079thatisthenumberofscientificworkers

whoareolderthantheaverageage(amongthescientificworkers)islargerthanthatyoungerthantheaverageageTheaverageageamongthescientificworkersis =4125years

1319 fornge2v+1m2v+1=0Forthecalculationofintegralsoftheform

make the change of variables that leads to the B-functionandexpressthelatterintermsoftheT-function

1320

1321M[X]=0D[X]=

1322 wheremj=M[Xj]

1323 where

14 POISSONrsquoSLAW

141p=1ndashendash01asymp0095

142 143p=1ndashendash1asymp063

144p=endash05asymp061145(1)095958(2)095963146091470143

148 14904

1410Sk=

1411 1412M[X] = D[X] = (log2)(log e)MN0ATπ Work out the differential

equationfortheaveragenumberofparticlesattheinstanttEquatetheaveragenumberofparticleswithhalftheinitialnumberTheresultingequationenablesone to find the probability of decay of a given particlemultiplying it by thenumberofparticleswegetM[X]

1413(a) (b)p=1ndashendashnndashnendashnasymp0673where

1414ExpressPn(k1k2hellipkmkm+1)intheform

where Inasmuchas andsisfinitethen


151p=00536152pbelow=01725pinside=04846pabove=03429153(a)1372sqm(b)0410515422measurements

155 156SeeTable122

TABLE122

157Easymp39mThe resulting transcendental equationmaybemore simplysolvedbyagraphicalmethod

158 159(a)0158700228000135(b)0317300455000271510pasymp00891511p=0251512(a)05196(b)012811513M[X]=3items1514Notlessthan30μ1515~86km1516(a)125mm(b)073mm1517

1518


161E(u)=q+peiuwhereq=1ndashp

162 wherepk+qk=1163E(u)=(q+peiu)nM[X]=npD[X]=npq

164 M[X]=aD[X]=a(1+a)165E(u)=expa(eiundash1)M[X]=D[X]=a

166

167E(u)= mk=k

168

169 wherev=u2hand

Integratebypartsandthenusetheformulas

1610

1611 Pass to polar coordinatesanduseoneoftheintegralrepresentationsoftheBesselfunction2

1612E(u)=exp[ixundasha |u]Byachangeofvariablesitisreducedtotheform

Theintegralinthisformulaiscomputedwiththeaidofthetheoryofresiduesforwhichitisnecessarytoconsidertheintegral

over a closed contour For positive u the integration is performed over thesemicircle(closedbyadiameter)intheupperhalf-planeandfornegativenoverasimilarsemicircleinthelowerhalf-plane

1613

1614μ2k=σ2k(2kndash1)μ2k+1=0

1615 (theCauchylaw)1616

Solvethiswiththeaidofthetheoryofresiduesconsiderseparatethecasesofpositiveandnegativevaluesofx

1617 P(X = k) = 2ndash k where k = 1 2 3 hellip Expand the characteristicfunction ina seriesofpowersof (l2)eiu anduse theanalytic representationofthe8-functiongivenintheintroductiontoSection11p49


171 172Denotingthediameterof thecirclebyDand the intervalbetween the

pointsbylweobtain

173p=015174

175Inbothcaseswegetthesameresultp1=p2=04176

177

178

179

1710



181

F(xy)=F1(x)F2(y)where

182 (a) A = 20 (b)

183f(xyz)=abcendash(ax+by+cz)184Thetrianglewithverticeshavingcoordinates

185(a)F(ij)=P(XltiYltj)=P(Xleindash1Ylejndash1)ForthevaluesofF(ij)seeTable123

TABLE123

(b)1ndashP(Xle6Yle1)=1ndash0887=0113

(c)M[X]=1947M[Y]=0504

186

187 188P=f(uvw)[f(uvw)+f(uwv)+f(vuw)+f(vwu)+f(wuv)+

f(wvu)]189P=F(a1b3)ndashF(a1b5)ndashF(a2b1)ndashF(a2b3)+F(a3b4)ndashF(a3b2)+

F(a4b2)ndashF(a4b4)+F(a5b5)ndashF(a5b1)1810P=andash3ndashandash6ndashandash9+andash121811

whereα=arccos(aR)β=arccos(bR)

1812

1813 1814Considertheexpectationsofthesquaresoftheexpressions

1815Makeuseofthereactionkxy=M[XY]ndash

18161817

1818

1819f(xy)=cosxcosyM[X]=M[Y]= ndash1

1820

1821

HintUsetheformulaP(AcupB)=P(A)+P(b)ndashP(AB)wheretheeventAmeansthattheneedlecrossesthesideaandBthatitcrossessideb


191192

193

194195

196

197

198P(k)=1ndashendashk22

199

1910 whereI0(x)istheBesselfunctionofanimaginaryargument

1911(a)P(XltY)= (b)P(Xlt0Ygt0)=

1912

1913

1914

1915

19161917

1918

wherep2=0196p3=0198p4=0148p5=0055q=0403

1919P= [Φ(k)]2

19201921

1922

192325(x1ndash10)2+36(x1ndash10)(x2ndash10)+36(x2ndash10)2=748461924

1925Theproblemhasnosolutionforngt12


201

202For|x|leR|y|leR

XandYareindependentsincef(xy)nefx(x)fy(y)

203δ(z)beingtheδ-function

204205

206fz(z)=[3(R2ndashz2)4R3]for|z|ltRf(xy|z)=1[π(R2ndashz2)]for|z|ltR207k=4fx(x)=2xendashx2(xge0)fy(y)=2yendashy

2(yge0)f(x|y)=fx(x)f(y|x)

=fy(y)M[X]=M[Y]= 2D[X]=D[Y]=1ndashπ4kxy=0208

209SinceM[X]=5M[Y]=ndash2σx=σσy=2σr=ndash08itfollowsthat(a)M[X|y]=5ndash082(y+2)=42ndash04yM[Y|x]=ndash2ndash08times2(xndash5)=6ndash16xσx|y=06σσy|x=12σ

2010

FortheindependenceofXandYitisnecessarythat

Thisconditionissatisfiedforb=0Inthiscase

2011

2012

2013M[X|y]=08y+149M[Y|x]=045xndash8625

2014

2015whereI0(x)istheBesselfunctionofzeroorderofanimaginaryargument

2016

2017

2018

2019



2114aπ212π(a2)213M[G]=41gD[G]=032g2

21421540πcm216M[y]=1217115m218a22219(nndash2)pq2(fornge3)

2110211111a218π21123π2113

2114

21152116n[1ndash(1ndashp)m]2117

2118

where istheprobabilitythatafterthefirstseriesofcyclesexactlykunitswillbedamagedatleastonce

2119

where forn=m=8(b)2mpforngt2m

2120

2121

212221230316g2124l3l2182125M[Z]=5aD[Z]=100a2+225b2ndash150ab

21262127

2128(a)267sqm(b)220sqm(c)10sqm2129

21302131

2132M[Z]=0D[Z]=2Δ2σ2

21332134


221222fy(y)=fx(ey)ey

223

224

225

226

227

228

229(b)ifagt0then

ifalt0then

(c)2210Foranoddn

forevenn

2211

22122213

2214

22162217

2218

2219

whereI0(z)istheBesselfunctionofzeroorderofimaginaryargument

2220

2221

2222Forrxy = 0Φ is uniformly distributed in the interval (0 2π) and the randomvariableTobeysaRayleighdistributionlaw

2223 f(s | t) is the probability density of a normal distribution withparameters

2224Thecharacteristicfunctionoftherandomvariable ifσ2=1 jisExj(t)=(1ndash 2t)ndash12 Then the characteristic function of the random variable

willbeEu(t)=(1ndash2t)ndashn2andtheprobabilitydensity

IftherandomvariablesXjhavethesamevarianceσ2and j=0thentherandomvariable

Consequently whereΨ(y)=y2nσ22225

2226

2227


231Makeuseofthefactthatforindependentrandomvariables

232Ez(u)=Exlx2hellipxn(uuhellipu)

233234

235236Ey(u)=(1+iu)ndash1mr=M[Yr]=(ndash1)rr

237Ey(u)=J0(au)where istheBesselfunctionoffirstkindofzeroorder

238

239

231023112312

2313M[X1X2X3]=02314M[X1X2X3X4]=k12k34+k13k24+k14k232315Fortheproofmakeuseoftheexpansionofthecharacteristicfunction

inaninfinitepowerseriesofu1u2hellipun2316Fortheproofusetheproperty

whereE(u1hellipun)isthecharacteristicfunctionofasystemofnormalrandomvariables

2317

24 CONVOLUTIONOFDISTRIBUTIONLAWS241

241242

243

where

244

245 The convolution of the normal distribution law with the uniformprobabilitylawhastheprobabilitydensity

Equatingtheexpectationandvariancefor fz(z)andfor theprobabilitydensity fprimez(z)ofthenormaldistributionlawweobtain

where

If =0thentherelativeerrorofsuchasubstitutionatthepointz=0is

TABLE124

246where c =a +b l =hk(h + k) (For solutionmake use of the characteristicfunctionsoftherandomvariablesXandY)

247

248249

whereI0(z)istheBesselfunctionofzeroorder

2412

2413Therequiredreserveresistanceis037middot =74kg2414

2415

2416

24172418SeeTable125

TABLE125

24192420TherandomvariableYhasbionomialdistribution2421Fz(n)=P(Zltn)=1ndash(n2nndash1)(n=12hellip)


251EQasymp9100cal252

253

254

255Easymp6666mEyasymp3860m256Eυ1asymp052msec257 For the assumed conditions the function V1 = ndash V cos q cannot be

linearized258σxasymp231mσyasymp143mσzasymp25m259σx=σyasymp866mσzasymp705m

25102511Eh=43m2512σzasymp10ndash62513Ehasymp1298m2514 The standard deviation of errors in determination of distance by the

formulausingthedataoftheradarstationisasymp2285m2515

2516

2517

2518 (a) By retaining the first two terms of the expansion in the TaylorseriesofthefunctionY=1Xweobtain asympndash02D[Y]asymp016(b)ByretainingthefirstthreetermsoftheexpansionintheTaylorseriesofthefunctionY=1Xweobtain asympndash100D[Y]asymp144

2519(a)Bytheexactformulas

(b)accordingtotheformulasofthelinearizationmethod

2520 (a) Measuring the height of the cone we get D[V] asymp 4π2 (b) bymeasuringthelengthofthegeneratorwegetD[V]asymp3577π2

2521199mg

2522

2523


261Anormaldistributionlawwithprincipalsemi-axesoftheunitellipsea=484mb=124mmakingc1theanglesα=19deg40primeand109deg40primewiththedeviationvectors

262Forγ=0adegeneratenormallaw(deviationvector) =50mForγ = 90deg a normal distribution lawwithprincipal semi-axesof theunitellipse a = c1 = 30m b = c2 = 40m coincidingwith the directions of thedeviationvectors

263Theprincipalsemi-axesa=12mb=11mmakeanglesof33degand123degwiththex-axis

264Theprincipalsemi-axesa=b=100m that is thetotaldispersioniscircular

265a=308mb=260mα=18deg15prime

266(a)(a)a=b=25 m(b)a=689mb=388mα=15deg

267Fromthesystemofequationsfortheconjugatesemi-diametersmandnm2+n2=a2+b2mn=ab(sinγ)wefindm=20mn=15mand

268|m|=732m|n|=681mε=74deg21prime269(a)f(xy)=117middot10ndash5expndash706middot10ndash2(0295x2ndash0610xy+13ly2)

(b)a=1265mb=538mα=12deg10prime2610a=880mb=257mα=39deg12prime2611Thedistributionlawisdefinedbytwoerrorvectors(Figure42)

FIGURE42

2612a=180kmb=739kmα=85deg36prime2613Totheerrorvectorsa1anda2oneshouldaddanothererrorvectora3

fora3 = β0 which gives at the pointC a unit ellipse of errors with principalsemi-axesa=412mb=197mmakingwith thedirectionof thebase theangles74deg20primeand164deg20prime

2614Eυ=21msecEq=0042rad2615a = 156mb = 139m the principal semi-axes directed along the

courseoftheship2616a = 640mb =c = 781m the semi-axis a is directed along the

courseoftheship2617

2618Theequationoftheunitellipsoidis

2619

2620p=ndash147middot107q=ndash89middot109φ=65deg45primeu1=4106u2=ndash622u3=ndash3484a=893b=570c=193cos(ax)=plusmn06179cos(ay)= 03528cos(az)= 07025

FIGURE43

FIGURE44

2621Ifwetakeasthex-axis(Figure43)thedirectionBK2andasthey-axisthedirectionperpendiculartoitthenbythelinearizationmethodwefindthreeerrorvectors

Fromthiswefind

2622 The error vectors a2 and a3 remain the same in magnitude anddirection as in the preceding problem The magnitude of the error vector a1causedby the error in the distanceD1 and its direction isdeterminedfromtheformulas(Figure44)

where



271Since

theoutcomeoftheexperimentforthefirsturnismorecertain272p=12273

thatistheuncertaintiesarepracticallythesame274

275SinceP(X=k)=p(1ndashp)kndash1then

Whenpdecreasesfrom1to0theentropyincreasesmonotonicallyfrom0toinfin276

277(a)loga(dndashc)(b)loga[ ](c)loga(ec)

278H[X]=loga(05 )

279whereσxandσyare thestandarddeviationsandr is thecorrelationcoefficientbetweenXandY

2710

where|k|isthedeterminantofthecovariancematrix2711Hx[Y]=H[Y]ndashH[X]+Hy[X]2712Theuniformdistributionlaw

2713Theexponentialdistributionlaw

27142715Thenormallaw

27162717loga1050andloga302718

where I(partφkpartxj) is the Jacobian of the transformation from (Y1Y2hellipYn) to(X1X2hellipXn)

2719 (a) The logarithm of the absolute value of the determinant |akj| (b)185decimalunit


281(a)5binaryunits(b)5binaryunits(c)3binaryunits282 For a number of coins satisfying the inequality 3k ndash 1 lt N le 3k k

weighingsarenecessaryFork=5onemayfindacounterfeitcoinif thetotalnumberofcoinsdoesnotexceed243

283I=500(ndash051log2051ndash031log2031ndash012log2012ndash006log2006)=815binaryunits

284Thefirstexperimentgivestheamountofinformation

andthesecondexperiment

285TheminimalnumberoftestsisthreeinthesequencesNo6No5andNo3 forexampleHintDetermine theamountof informationgivenbyeachtest and select as the first test one of those that maximizes the amount ofinformationSimilarlyselectthenumbersofsuccessivetestsuntiltheentropyofthesystemvanishesTocomputetheamountof informationusetheanswertotheprecedingproblem

286whereP(aj)=P(Ai)ifthecodeajcorrespondstothesymbolAiofthealphabetForcodeNo1

ForcodeNo2

287Foramoreefficientcodethesymbolsofthecodewiththesameserialnumbersarrangedintheorderoftheirincreasingdurationsshouldcorrespondtothe symbols of the alphabet arranged in theorder of decreasingprobabilities thatisthesymbolsdcbandaof thecodeshouldcorrespondtothesymbolsA1A4A3andA2Theefficiencyofsuchacodeis

288

289(a)SeeTable126TABLE126

(b)SeeTable127

TABLE127

(c)SeeTable128

TABLE128

Theefficienciesofthecodesarerespectively

(a)

(b)P(1)= =0615P(0)=0385lσ=1ndash0962=00382810(a)P(1)=08P(0)=02la=1ndash0722=0278

(c)P(1)= =0528P(0)=0472lB=1ndash09977=000232811(a)SeeTable129and130

TABLE129

TABLE130

(b) The efficiencies of the codes are 0890 and 0993 respectively (c) Theredundanciesofthecodesare0109and00007respectively


2813Use the fact that thecodednotationof the letterAjwill consistofkjsymbols

2814Intheabsenceofnoisetheamountofinformationistheentropyoftheinputcommunicationsystem

InthepresenceofnoiseI=0919binaryunititdecreasesbyanamountequaltothemagnitudeoftheaverageconditionalentropynamely

where

2815IfthenoiseisabsentI=H1=log2mwhenthenoiseispresentI=H1ndashH2=log2m+plog2p+qlog2q(mndash1)

2816

where

VI THELIMITTHEOREMS


291

292ItisprovedinthesamemannerasoneprovesChebyshevrsquosinequalityFortheproofmakeuseoftheobviousinequality

whereΩisthesetofallxsatisfyingthecondition

293 Using arguments analogous to those in the proof of the Chebyshevinequalityoneobtainsachainofinequalities

294UsetheChebyshevinequalityandnotethat =m+1andM[X2]=(m+1)(m+2)hence

295DenotingbyXntherandomnumberofoccurrencesoftheeventAinn

experiments we have P(|Xn ndash 500| lt 100) gt 1 ndash (2501002) = 0975Consequentlyallquestionsmaybeansweredldquoyesrdquo

296 The random variables Xk are mutually independent and have equalexpectations k=0andvariancesD[Xk]=1whichprovethattheconditionsoftheChebyshevtheoremaresatisfied

297Forslt12sinceinthiscase

298

whichprovestheapplicabilityofthelawoflargenumbers299(a)Notsatisfiedsince

(b)satisfiedsince

(c)notsatisfiedsince

2910Applicablesincetheinequality

wherecistheupperboundofD[Xk]forallk=12hellipnholdsforkijlt0Therelation

followsfromtheinequality2911Toprovethisitsufficestoestimate

where

Replacingallσkbytheirmaximalvaluebweobtain

henceitfollowsimmediatelythat

2912 Applicable since all the assumptions of Khinchinrsquos theorem aresatisfied

2913Consider

whereσiisthestandarddeviationoftherandomvariableXiSincerijrarr0for|indashj|rarrinfinthenforanyεgt0onemayindicateanNsuchthattheinequality|rij|ltε

holds for all |i ndash j| gt N This means that in the matrix containing n2elementsatmostNnelementsexceedε (theseelementsare replacedbyunity)andtherestarelessthanεFromtheprecedingfactsweinfertheinequality

thereforelimnrarrinfinD[zn]=0thisprovesthetheorem2914Thelawoflargenumberscannotbeappliedsincetheseries

definingM[Xi]isnotabsolutelyconvergent


301302P(70lemlt86)=0927303(a)P(mge20)=05(b)P(mlt28)=09772

(c)P(14lemlt26)=08664304InthelimitingequalityofthedeMoivre-Laplacetheoremset

andthenmakeuseoftheintegralrepresentationsofthefunctionsΦ(x)andΦ(x)305 Because the probability of the event is unknown the variance of the

numberof occurrencesof the event shouldbe taken asmaximal that ispq =025Inthiscase(a)nasymp250000(b)n=16600

306 In the problems inwhich the upper limit of the permitted number ofoccurrencesisequaltothenumberofexperimentsperformedbturnsouttobesolargethatΦ(b)asymp1Inthiscasenasymp108

307nasymp65308p=0943309675

3010 dxmaybeconsideredasthemomentofsecondorderofarandomvariableuniformlydistributedovertheinterval[01]thenitsstatisticalanalogdeterminedbyaMonte-Carlomethodwillbe whereXkarerandomnumbersontheinterval[01]WiththeaidofLyapunovrsquostheoremwefindthatP(|J1000ndashJ|lt001)=071

3011nasymp155middot106Set whereXkare randomnumbersfromtheinterval(0π2)

3012(a)Sincethedifference

fromthepointofviewofthelawoflargenumbersbothmethodsleadtocorrectresults (b) in the first case 9750 experiments will be necessary and in thesecondcase4500experiments

3013(a)3100(b)15003014Inallthreecasesthelimitingcharacteristicfunctionequalsendashu22

3015



311Denotingbyf(x1x2|t1t2)thedistributionlawofsecondorderfortherandomfunctionX(t)bythedefinitionofKx(t1t2)wehave

ApplyingtheSchwarrzinequalityweget

which is equivalent to the first inequality To prove the second inequality itsuffices to consider the evident relation

312Theproofissimilartotheprecedingone

313Itfollowsfromthedefinitionofthecorrelationfunction314Since wherec isanonrandomconstantand

nisthenumberofstepsduringtimetwehaveD[X(t)]=M[nσ2]=λtσ2315ThecorrelationfunctionKx(τ)istheprobabilitythatanevennumberof

signchangeswilloccurduringtimeτminustheprobabilityofanoddnumberofsignchangesthatis

316SinceM[X(t)X(t+τ)]ne0only if (t t + τ) is contained in an intervalbetweenconsecutiveintegersandsincetheprobabilityofthiseventis0if|τ|gt0and(1ndash|τ|)if|τ|le1wehavefor|τ|le1

Consequently

317LettingΘ1=Θ(t1)Θ2=Θ(t1+τ)for theconditionaldistributionlawweget

where f(θ1θ2) is thenormaldistribution lawof a systemof randomvariableswithcorrelationmatrix

Substitutingthedatafromtheassumptionoftheproblemweget

318 Denoting the heel angles at instants t and t + τ by Θ1 and Θ2respectively and their distribution law by f(θ1 θ2) for the conditionaldistributionlawoftheheelangleattheinstantofsecondmeasurementweget

Therequiredprobabilityis

319DenotingX1=Θ(t)X2=Θ(t)X3=Θ(t+τ0)thecorrelationmatrixofthesystemX1X2X3becomes

whichafternumericalsubstitutionbecomes

Determining the conditional distribution law according to the distribution lawf(x1x2x3)

weobtainfortherequiredprobability

3110

3111

3112TheprobabilitythattheintervalTwillliebetweenτandτ+dτistheprobabilitythattherewillbenpoints in theinterval(0τ)andonepoint in theinterval(ττ+dτ)Sincebyassumptiontheseeventsareindependentwehave

thatis

3113


321Since hasnodiscontinuityatτ=0

322323Usingthedefinitionofamutualcorrelationfunctionweget

324 Since any derivative of Kx(τ) is continuous at zero X(t) may bedifferentiatedanynumberoftimes

325Twicesince(d2dτ2)Kx(τ)|τ=0and(d4dτ4)Kx(τ)|τ=0exist(d5dτ5)Kx(τ)hasadiscontinuityatzero

326Only the firstderivativeexists since (d2dτ2)Kx(τ) exists for τ=0and(d3dτ3)Kx(τ)hasadiscontinuityatthispoint

3273283293210Thedistributionf(υ)isnormalwithvariance and

=0P=030853211

3212

32133214

3215Since ifwe let t2=t1 = t pass to new variables of integration and perform the integration weobtain

3216 Solving the problem as we did 3215 after transformation of thedoubleintegralweget

32173218D[Y(20)]=1360cm23219

3220

3221SincethevarianceD[θ(t)]issmallsinθasympθ

whichaftersubstitutionofnumericalvaluesleadstoσΔυ=186msec3222Using thedefinitionof thecorrelation functionas theexpectationof

the product of the deviations of the ordinates of a random function and theformulasforthemomentsofnormalrandomvariablesweobtain

32233224Ky(τ)=endashα

2τ2[1+2α2(1ndash2α2τ2)]

32253226

3227Itdoesnotexist3228(a)Stationary(b)nonstationary3229

Fort=1hourσyasymp15km3230D[α(t)]asympa1tD[β(t)]asympb1t

and arethenormalizedcorrelationfunctions and

3231

where


331332D[V(t)]=025cm2sec2333Thenumberofpassages(goingup)beyondthelevela=25degequalsthe

numberofpassagesgoingdownbeyond the levela=ndash25degconsequently therequirednumberofpassages

334335Startingwith 336Theproblemreducestothedeterminationofthenumberofpassagesof

therandomfunction beyond the level (goingup)andndash(goingdown)

Answer

337Sincetheradiusofcurvatureis thesensitiveelementreaches

astopwhen leavesthelimitsofthestripplusmnυR0whichleadsto

338For ge545m

339

3310Denotingbyf(xx1x2)theprobabilitydensityofthesystemofnormal

variablesX(t) and wegettherequiredprobabilitydensity

Consideringthatthecorrelationmatrixhastheform

wefindafterintegrationthat

33113312Therequirednumberequalsthenumberofpassages(frombothsides)

beyondthezerolevelconsequently

3313

where

are the cofactors of the determinant Δ2 and kjl are included in the answer toProblem3314

3314 is the probability density p of sign changes for ζx and ζy in thevicinityofthepointwithcoordinatesxyThesearerelatedasfollows

TheprobabilitypdxdycanbecomputedifoneconsidersthatK(ζη)uniquelydefines the distribution law of Performingthecomputationsweobtain

where


341

342343Denoting

wehave

344345

346

347Solvingthisproblemaswedid343weget

348349TwoderivativessinceSx(ω)decreasesas1ω2whenωincreases

34103411

Consequently forω = 0 therewill always be an extremum If forω=0 theexpressionbetweenbrackets isnegative thesignof thederivativeat thispointchanges fromplus tominus therewill beonemaximumat this point andnoothermaximaThus the condition for nomaxima except at the origin isα2gt3β2For

thatisS(ω)alsocanhaveonlyonemaximumat theoriginTherefore ifα2ge3β2thenthereexistsonemaximumattheoriginifα2lt3β2therewillbeoneminimumattheoriginandtwomaximaatthepoints

3412Since

then

3413Since

then

3414Since

theFouriertransformleadsto

3415

34163417Since

thentheFourierinversionleadsto

where

3418Since then

3419SinceKΔ(τ)=Kψ(τ)Kθ(τ)theFouriertransformleadsto

where

3420Applyingthegeneralformula

andtheresultsofProblem3417weget

34213422

3423

whereSφ(ω)=S1(ω)Sθ(ω)=S2(ω)Sψ(ω)=S3(ω)

andalltheintegralsmaybecomputedinafiniteformBecausethefinalresultiscumbersome in the present case it is preferable to use numerical integrationmethods

3424Since then

hasonemaximumforω=0

3425

where

andj0istheintensityofphotocurrentcreatedwhenoneholecoincideswiththeapertureofthediaphragm

35 COMPUTATIONOFPROBABILITYCHARACTERISTICSOFRANDOMFUNCTIONSATTHEOUTPUTOFDYNAMICALSYSTEMS

351Y(t)isastationaryfunctionconsequently

whichafteraFourierinversionyields

352 Since Y(t) is stationary finding the expectation of both sides of theequationweobtainthat Thespectraldensityis

whichafterintegrationbetweeninfinitelimitsgives

353

where

354 Since by the assumption of the problem α(t) can be consideredstationary

whereSu(ω) isobtainedas inProblem353 IntegratingSα(ω)between infinite

limitswiththeaidofresiduesweget =213middot10ndash6rad2σα=146middot10ndash3rad

355

where Applying a FourierinversiontoSy(ω)weget

356

where

357 358Nosincetherootsofthecharacteristicequationhavepositiverealparts

andconsequentlythesystemdescribedbytheequationisnonstationary359Sinceζc(t)isstationaryitfollowsthat

3510Lettingω0=na=3middot10ndash4g2wegetD[ε(t)]=D[ζc(t)]whereD[ζc(t)]ismentionedintheanswertoProblem359SubstitutingthenumericaldatawegetD[ε(t)]=006513σε=0255

3511 The formula is a consequence of the general formula given in theintroduction

3512Lettingω0=kweobtainD[Θ(t)]=D[ζc(t)]whereD[ζc(t)]isgivenintheanswertoProblem359

35133514Theindependentparticularintegralsofthehomogeneousequationare

endashtendash7ttheweightfunctionisp(t)=(16)(endashtndashendash7t

3515

wheretheminussigninthelowerlimitsofintegrationmeansthatthepoint0isincludedinthedomainofintegration

35163517 =constwhosevaluemaybetakenzerobyaproperchoiceofthe

origin

3518ReplacingX(t) by its spectral decompositionwe obtain the spectraldecompositionof

where Fromthisitfollowsthat

whichafterwesubstitutetheexpressionforSx(ω)andintegratewiththeaidofresiduesgivesthefinalresultinthefiniteform

3519

3520

3521

3522

where y1(t) hellip yn(t) are the independent particular integrals of thecorrespondinghomogeneousequation

andAjlarethecofactorsofthisdeterminant3523Sincethesolutionofthesystemleadsto

and

then

3524

3525D[Y1(05)]=001078D[Y2(05)]=0001503526SinceY(t)andZ(t)canbeassumedstationary


3527Anormallawwithparameters =0σy=0783528

3529 To find the asymmetry and the excess one should determine themoments of Y(t) up to and including the fourth To find these moments it isnecessarytofindtheexpectations

forthedeterminationofwhichoneshouldtakethederivativesofcorrespondingordersof thecharacteristic functionof thesystemofnormal randomvariablesForexample

where ||kjl|| is the correlation matrix of the system of random variablesX(t1)X(t1)X(t2)X(t2)

Substitutingtheobtainedexpressionsinthegeneralformulasformomentsofthesolutionofadifferentialequationweget

3530Forτge0weshallhave

andforτle0


361 Determining Kx(τ) as a correlation function of a sum of correlatedrandomfunctionsandapplyingtotheresultingequalityaFourierinversionweget

362Sxz(ω)=iω[Su(ω)+Svu(ω)]363L(iω)=iωendashiωτD[ε(t)]=0364

where

365

where

366

367

where

where

368L(iω)=endashατ369L(iω)=endashτ[iωτ+(1+τ)]3610

3611

where

3612

where

3613 3614

where

3615Therequiredquantityischaracterizedbythestandarddeviationoftheerror of the optimal dynamical system of 167 0738 00627 msecrespectively

3616 where

whichgivesforσεthevalues162082900846msec3617

where

3618

where

3619ThegeneralformulaforL(iω)isthesameasintheprecedingproblemexceptthat

3620l(τ)=δ(τ)D[ε(t)]=03621Forthefirstsystem

theconstantsλ1λ2λ3andλ4aredeterminedfromthesystem

which has the solutions λ1 = ndash 00018 λ2 = 0000011 λ3 = ndash 00106 λ4 =00036ThevariancefortheoptimalsystemoffirsttypeisD[ε(t)]=0135middot10ndash4ForthesecondsystemtheformofL(iω)remainsthesamebutλ1=λ2=0andλ3λ4aredeterminedfromthesystem

whichleadstoλ3=ndash00136λ4=00023Thevarianceforthissystemis

36223623

3624

3625

3626


371

where

Ei(x)denotestheintegralexponentialfunction

372Since

wehaveω1=2απω2=α

areindependentofα373

374P=05andareindependentofαβ

375376Thephaseisuniformlydistributedovertheinterval[02π]377

378

379Sincek(τ)=endashα|τ|(1+α|τ|)k(2)=0982

then

3710Since

thefollowingformulaisuseful

3711 3712 The required average number of passages equals the probability of

occurrenceofonepassageperunittime

371300424αsecndash13714

where

3715


38 MARKOVCHAINS

381Itfollowsfromtheequality 382p(3)=Rprimep(0)where

383StatesQ1meansthatallcompetitionsarewonQ2meansthatthereisonetieQ3meansthatasportsmaniseliminatedfromthecompetitionsBythe

Perron formula

384 StatesQ1means that thedevice is ingood repairQ2means that theblockingsystemisoutoforderQ3meansthatthedevicedoesnotoperate

385ThestateQj(j=0123)meansthatjmembersofateamparticipateincompetitionsForiltk =0(ik=0123)

where

386MakeuseofPerronrsquosformulaforsingleeigenvalues

ForigtkAki(λ)=0

Forkgti

387UsePerronrsquosformulawhentheeigenvalueλ=phasmultiplicitymandtheeigenvalueλ=1isnotmultiple

ForigtkAki(λ)=0

Forkgtiknem

388 The stateQj means that there are j white balls in the urn after thedrawingsForjgtipij=0forigej

Theeigenvaluesλ0=1 (k=12hellipm)arenotmultipleThe

transposedmatrix isuppertriangulartheprobabilities aredeterminedbytheformulasfromthehypothesisofProblem386ForN=6m=3

389StateQjmeansthatthemaximalnumberofpointsisN+jpii=impij=0forigtjpij=1mforiltj(seeExample381)

3810ThestateQjmeansthat jcylinders (j=01hellipm) remainedon thesegmentoflengthLTheprobabilitythattheballhitsacylinderisjαwhere

forinejandinejndash1(ij=01hellipm)Theeigenvaluesλk=1ndashkα(k=01hellip

m) =0foriltkForigek

ByPerronrsquosformulaforigekwehave

3811StateQj(j=12hellipm)meansthattheselectedpointsarelocatedinjpartsoftheregionDpjj=jmpjj+1=1ndashjmTheeigenvaluesλr=rm(r=1

2hellipm)From H=HJitfollowsthatfrom and it follows that

for i gtk andforilek

(foranothersolutionseeProblem3810)3812Setε=e2πimThen

where

3813Qirepresentsthestateinwhichtheparticleisatpointxi

Thematrixequation isequivalenttotheequations

where

SinceRi(ξ)isapolynomialtheeigenvaluesλi=1ndash2im(i=01hellipm)From

itfollowsthat Letting

wefind theelements of thematricesH=H ndash 1 are givenby theexpression

Theprobabilities aretheelementsofthematrix

3814Qj describes a state in which the container of the vendingmachinecontainsjnickels

Theeigenvaluesare

where

TheconstantsCjaredeterminedfromthecondition

3815StateQ1meanshittingthetargetandQ2meansafailure

Theeigenvaluesareλ1=1λ2=αndashβBytheLagrange-Sylvesterformulaforλ2ne1weget

Ifλ2=1then

3816From (j=12hellipm) itfollowsthat =1m(j=12hellipm)

3817Qjdescribesthestateinwhichtheurncontainsjwhiteballs

Thechainisirreducibleandnonperiodic Fromthesystem

weget

3818Qjdescribesthestateinwhichtheparticleislocatedatthemidpointofthejthintervalofthesegment

The chain is irreducible and nonperiodic The probabilities can bedeterminedfromthesystem

Then

Forp=q =1mandforpneq

Theprobabilities canalsobeobtainedfrom asnrarrinfin(seeProblem3814)

3819ThechainisirreducibleandnonperiodicFromthesystem

itfollowsthat

Since

thereisanonzerosolutionWealsohave

thatisthechainisergodic


itfollowsthat Wealsohave

consequentlythechainisergodic

thatis


itfollowsthat

Theseries

is divergent that is the chain is nonergodic This is a null-regular chain forwhich =0(ik=12hellip)

3822QjmeansthattheparticleislocatedatthepointwithcoordinatejΔ(j=12hellip)

The chain is irreducible and nonperiodic From the system uipij = ujfollowsthatuk=(αβ)kndash1u1(k=12hellip)For(αβ)lt1wehave

andconsequentlythechainisergodic

thatis

Ifαβge1theMarkovchainisnull-regular =0(jk=12hellip)3823SinceWinfin=0 =1(j=s+1s+2hellipm)3824Fromthesystem

weobtain

3825QjrepresentsthestateinwhichplayerAhasjdollars(j=01hellipm)p00=1pmm=1pjj+1=ppjj ndash 1=q (j=01hellipmndash1)Theprobabilities

ofruinofplayerAaredeterminedfromthesystem

Setting =andashb(qp)jwefindforpneqthat

andforp=qthat =1ndashjm(j=12hellipmndash1)TheprobabilitiesofruinofB are Another solution of this problem may beobtainedfromtheexpressionfor asnrarrinfin(seeExample382)

3826H = ||hjk|| = ||ε(j ndash 1)(k ndash 1)|| where ε = e2πim Then H =H ||δjkλk||whereλk=εkndash1(k=12hellipm)Since|λk|=1theperiodκ=m

thatis =1ifn+jndashkisdivisiblebymand =0otherwise(jk=12hellipm) =1ifr+jndashkisdivisiblebymand =0otherwise(r=01hellipmndash1)

3827

whereε =e2πi3Theperiodκ = 3For jk = 2 3 4 =1 ifn + j ndashk isdivisibleby3and =0otherwiseBythePerronformula

3828Thechainisirreducibleandperiodicwithperiodκ=2Thefirstgroupconsistsof stateswithoddnumbers and the second thosewithevennumbersThenlimnrarrinfin =pkandlimnrarrinfin =0ifj+kisanevennumberandlimnrarrinfin =0limnrarrinfin =pkifj+kisanoddnumberThemeanlimitingabsoluteprobabilities =12m(k=12hellip2m)aredeterminedfromtheequality

3829Qjdescribesthestateinwhichtheparticleisatpointxj(j=01hellipm)p01=1pmmndash1=1pjj+1=pjjndash1=q(j=12hellipmndash1)Thechainisirreducible and periodic with period

(k=12hellipmndash1)Forpne

qwehave

Forp=qwehave (k=12hellipmndash1)


391

392 393

where

where istheexpectedflighttimeoftheelectron

394 395

396Solvingthefirstsystemofequations

forinitialconditionsPik(0)=δikbyinductionfromPik+1(t)toPik(t)toPik(t)weobtain

397Forλ=μtheinequality

givesm=4398Thesystemofequationsforthelimitingprobabilitiespn

hasthesolutions

wherep0isdeterminedbythecondition pn=1Theexpectednumberofmachinesinthewaitinglineis

399Thesystemofequationsforthelimitingprobabilitiespnis

andithasthesolutions

theexpectednumberofmachinesinthewaitinglineforrepairsis

3910Theprobabilitythatthecomputerrunsisthelimitingprobabilitythatthere areno calls for service in the systemp0 =endash λμwhereμ is the averagenumberofrepairsperhourTheexpectedefficiencyresultingfromapplicationofmorereliableelementsduring1000hoursofoperationis

3911(a)Thesystemofequationsforthelimitingprobabilities

hasthesolutions

where p0 is the probability that all devices need no service and can bedeterminedfromthecondition

withtheconditionthatλltnμ

wherePk(Tltt)istheprobabilitythatthewaitingtimeinthelineislongerthantiftherearekcallsforserviceinthesystem

Substitutingthisvalueweget

sincepkpn=(λnμ)kndashnchangingtheorderofsummationweobtainasaresult

andsincepnp=1ndash(λnμ)thenF(t)=1ndashpendash(nμndashλ)t(fortge0)

3912ApplytheformulasofProblem3911 =2115hours3913Selectnsothatpendash(nμndashλ)lt001n(seeProblem3911)3914(a)Thesystemofequationsforthelimitingprobabilities

wherel=n+mhasthesolutions

wherep0istheprobabilitythattherearenocallsforserviceinthesystem

(b)theprobabilityofrefusal

(c)theprobabilitythatalldevicesarebusyis

where

3915

3916Thesystemofequationsforthelimitingprobabilities

hasthesolutions

3917ThesystemofequationsfortheprobabilitiesPn(t)

forinitialconditionsPn(0)=δnlhasthesolutionPn(t)=endashλt(1ndashendashλt)nndash1

3918Thesystemsofequations

forinitialconditionsPn(0)=δn1issolvedwiththeaidofthegeneratingfunctionG(tu)= G(tu)satisfiesthedifferentialequation

withtheinitialconditionG(0u)=uIthasthesolution

where

thusitfollowsthat

3919Thesystemofequations

withtheinitialconditionPn(0)=δn0hasthesolutionsP0(t)=(1+at)ndash1a


401

402

theremainingbjl=0403U(t)equivU1(t)isthecomponentofatwo-dimensionalMarkovprocessfor

whicha1=x2

404aj(tx1hellipxn)=φj(tx1hellipxn)bjl=ψjl(tx1hellipxn)405TheMarkovprocesshasr+ndimensions

theotherbjl=0here 407

where ξ1(t) and ξ2(t) are mutually independent random functions with thepropertyofldquowhitenoiserdquo

408

wherecisdeterminedfromtheconditionsofnormalizationForφ(u)=β2u3

409

wherecisdeterminedfromthecondition f(y)dy=14010 SettingU1 = ζ(t)U2 =U1 ndashU forU2 we find an equation that is

independentofU1TheKolmogorovequationforU2willbe

anditsstationarysolutionis

where c is determined from the condition of normalization The requiredprobabilitydensityf(y)istheconvolutionoff(y2)andthenormaldistributionlawwithzeroexpectationIntheparticularcase

where

40114012TheKolmogorovequationforU=expndashaVhastheform

Thestationarysolutionis

where

(compareStratonovich1961p243)4013

where

4014TheKolmogorovequationis

theequationforthecharacteristicfunctionE(τz)is


4016SettingU1(t)=U(t)=U2(t)= wefind that thecoefficientsoftheKolmogorovequationare

where

40174018

where

and arethegeneralizedLaguerrepolynomials4019

whereDa(x) isanevensolutionof theWeberequation2 (theparaboliccylinderfunction)

ajisarootoftheequationDa(β)=0τ1=aτ

4020

where

and aretheevenandoddsolutionsoftheWeberequation2

ajistherootoftheequationVaj(β)=0 =ajndash05τ1=ατ



4111058m412(a)81487sqm(b)92186sqm413 =42473msec =884msec414 =33msec =307msec415 =40485sqm =133sqm416ForP(A)=05Dmax=frac14n

417

418

419

4110

41114112 whereλisanarbitrarynumber4113

thevaluesofknbeinggiveninTable23

4114 =4831m =5331m =1075m =1250m4115

where

andangleαisdeterminedfromtheequation

4116 =23m =107m4117

First showthat theprobabilitydensityof therandomvariable isdeterminedbytheformula



4120 and areunbiasedestimatesofthevariance

thatis (seeTable134)foranyngt2TABLE134


421(9236m10764m)

422 (11553m11657m)423055034424(a) =1057m =205m(b)026(c)0035425(5249sec5751sec)(1523sec1928sec)426(8676msec8730msec)427Notlessthan11measurements

428(24846m25154m)(1307m2949m)429(476110ndash10480510ndash10) =4783ndashl0ndash104210(a)(42075msec42865msec)(669msec1270msec)(b)

0610764211Notlessthanthreerangefinders4212Notlessthan15measurements42130440550710914214SeeTable135

TABLE135

4215 =425hours(27070hours77982hours)4216(41021hours103656hours)4217(5075hours8514hours)4218(01230459)4219(03030503)(02760534)4220(00000149)(00000206)(00000369)4221FormarksmanA(01280872)formarksmanB(03690631)4222(115324)4223(37214020)4224(046)4225

4226


431 =0928 =2172k=4 =0705Thedeviation isinsignificant thehypothesisonagreementoftheobservationswiththePoissondistributionlawisnotcontradicted

432 =154 = 7953 k = 6 = 0246 The deviation isinsignificant

433 =5p=05 =3156k=9 =0944Thehypothesisthatateachshottheprobabilityofhittingisthesameisnotdisproved

434 = 1032 k = 7 = 0176 The deviations areinsignificant

435Dhyp = 01068 λhyp = 1068 P(λhyp) = 0202Dbin = 01401 λbin =1401 P(λhyp) = 0039 The hypothesis that the observations agree with ahypergeometricdistributionlawisnotdisprovedthedeviationofthestatisticaldistribution from the binomial is significant and the hypothesis about thebinomialdistributionshouldberejected

436 =118g =4691gk=2 =116 =0568Thehypothesisthattheobservationsobeyanormaldistributionisnotdisproved

437 =2285 =6394k=6 =5939 =0436Thehypothesisthatthestatisticaldistributionagreeswithanormaldistributionisnotdisprovedsincethedeviationsareinsignificant

438M[Z]=45D[Z]=825whereZisarandomdigit

Thehypothesisthatthestatisticaldistributionagreeswithanormaldistributionisnotdisproved

439 = 5012 k = 9 = 0831 The deviations areinsignificant the hypothesis that the first 800decimals of the numberπ agreewithauniformdistributionlawisnotdisproved

4310D0=00138λ=03903P(λ)=0998Thehypothesis that the first800decimalsofπobeyauniformdistributionlawisnotdisproved

4311 = 4 k = 9 = 091 The hypothesis that theobservationsobeyauniformdistributionlawisnotrejected

4312 D0 = 0041 λ = 05021 P(λ) = 0963 The hypothesis that the

observations agree with a uniform distribution is not rejected since thedeviationsareinsignificant

4313 = 249 k = 9 = 00034 The deviations aresignificant the hypothesis that the experimental data agree with a uniformdistribution should be rejected The results of the computations contain asystematicerror

4314 =875 =1685 =1186kH=5 =00398an

estimate of is obtained for the parameter δ of theSimpson distribution law = 1706 kc = 5 = 000402 ThehypothesisthattheobservationsagreewiththeSimpsondistributionisrejectedandthehypothesisthattheyagreewithanormaldistributionmaybeconsiderednotrejected

4315 x = log y = ndash01312 = 03412 = 05841 n = 9 k = 6 = 0890 The hypothesis that the experimental data obey a

logarithmically normal distribution law is not disproved (the deviations areinsignificant)

4316

wherevistherootoftheequation

forv=12wehaveT(v)=04200forv=13T(v)=04241

ThehypothesisthatXistheabsolutevalueofanormallydistributedvariableisnotdisproved

4317 = 8746 = 2471 = 8002 = 9490 gt 500 kH = 7

The probability density Ψ(x) for the convolution of anormalanduniformdistributionhastheform

kφ = 6 = 0814 The hypothesis that the experimental dataobey a normal distribution law is disproved The hypothesis that theexperimental data agree with the convolution of a normal distribution and auniformoneisnotcontradicted

4318 =5013 =273k=8 =095The hypothesis that the observations agreewith aRayleigh distribution is notcontradicted

4319 =5086 =1237 =295kH=7 =0888Theparameter foraMaxwelldistributionisdeterminedfromtheformula

The observations fit a Maxwell distribution better than they fit a normaldistribution

4320 = 8715 hours = 0001148 k = 8 = 4495 =0808 The hypothesis that the observations agree with an exponentialdistributionlawisnotdisproved(thedeviationsareinsignificant)

4321 = 3945 hours = 2281 hours = 05782 = 1789 =08893 = 1344 k = 7 = 00629 The hypothesis on theagreementoftheobservationswithaWeibulldistributionisnotdisproved

4322Thearctandistributionlawis

ThehypothesisthatthestatisticaldistributionofvariablezagreeswithaCauchydistribution and consequently that of the variableYwith a normal one is notdisproved

4323Thearcsinedistributionfunction

The hypothesis that the pendulum performs harmonic oscillations is notdisproved

4324 =01211k=2 =1629 =059Thedeviationsareinsignificantthehypothesisthattheobservedvaluesofqiobeyachi-squaredistributionwithkprime=19degreesoffreedomandconsequentlythehypothesisonthehomogeneityoftheseriesofvariancesarenotdisprovedHintThevaluesofqishouldbearrangedintheirincreasingorderanddividedintointervalssothateachintervalcontainsatleastfivevaluesqi

4325

The hypothesis that the observed values obey a Studentrsquos distribution andconsequently the hypothesis that the observed values of xi obey a normaldistributionlawarenotrejected

4326 =1153 =2143 =1020kH=10 =043

=2046 =6137102 =02079 =ndash00912ThedistributionfunctionforaCharlier-Aseriesis

where

The hypotheses on the agreement of the observations with the normaldistributionandadistributionspecifiedbyaCharlier-Aseriesarenotdisprovedand the latter does not improve the agreement of the observations with thetheoreticaldistributionlaw

4327 = ndash22112 = 1560102 = ndash006961 = 03406 ThedistributionfunctionforaCharlier-Aseriesis

where

ThedeviationsaresignificantThehypothesisthattheobservationsagreewithawithadistributionspecifiedbyaCharlier-Aseriesisdisproved

4328 = 2048 k = 2 = 0001 The deviations aresignificant The hypothesis on the independence of the character of thedimensionsonthenumberof the lot is rejectedAsystematicunderestimateofdimensionsischaracteristicforthesecondlot


441 =0609+01242EM00=03896M11=000001156 =1464

=05704 =00000169442 =0679+0124E =1450 =05639 =000001672The

coincidencewiththeresultsofProblem441isfullysatisfactoryTheaccuracyoftheresultinProblem442ishigherthaninProblem441sinceinsolving441alargenumberofcomputationswereperformedandamongthemthereoccurredsubtractionofapproximatelyequalnumbers

443

444wherex=30tndash1or

445

446

wherePkl6arethetabulatedvaluesoftheChebyshevpolynomialsForalineardependence = 03048 for α = 090 we have 02362 lt σ lt 04380 For adependence of third degree = 01212 forα = 090we have 00924 ltσ lt01800

447 =2107+5954x =290 =00889 =ndash02041Theconfidenceintervalsforakforα=090are143ltaolt279575lta1lt616

=2900ndash04082x+00889x2Theconfidencelimitsfory=F(x)forα=090aregiveninTable136

TABLE136

448 =03548+006574x+000130x2 =00147 =00106 =000156

449

forα = 095 we have 1065 lt a0 lt 1172 8831 lt a1 lt 9115 = ndash00854Theconfidencelimitsfory=F(x)ifα=095aregiveninTable137

TABLE137

4410U=1008endash03127t8997ltU0lt112902935ltalt03319

44114412

4413φprime=62degischosenaccordingtotheformulay=aprimesin(ωtndashφprime)where

4414

4415 =ndash3924+1306x|εmax|=141

45 STATISTICALMETHODSOFQUALITYCONTROL

451Fora single sampleα=00323β=00190 for adouble sampleα=00067β=00100Theaverageexpenditureofitemsfor100lotsinthecaseofadoublesampleis483615+516430=2275itemsTheexpenditurefor100lots in the case of single sampling is 2200 itemsThe expenditure of items isalmostthesamebutinthecaseofdoublesamplingtheprobabilitiesoferrorsinαandβareconsiderablysmallerA=3038B=001963logA=14825logB=ndash17069Foragoodlotifp=0nmin=13logγ(120)(120)=ndash16288logγ(130)=ndash17771Foradefectivelotwhenp=lnmin=2logγ(11)=08451logγ(22)=19590

452Forasinglesampleα=0049β=0009foradoublesampleα=0046

β=0008A=198B=001053h1=ndash3758h2=2424h3=002915M[n|p0] = 2442M[n |p1] = 1136M[n]max = 3219 For 100 lots in the case ofdoublesamplingtheaverageexpenditureofitemsis351middot220+649middot440=36278 items in the case of single sampling the average expenditure is 41 000items In thecaseofsequentialanalysis theaverageexpenditure for100goodlotsisnotgreaterthan24420items

453 The normal distribution is applicable α = 00023 β = 00307 A =4159B=003077h1=ndash4295h2=7439h3=01452Foragoodlotifp=0nmin=30foradefectivelotifp=1nmin=9M[n|010]=9452M[n|020]=1289M[n]max=2574c=2153P(nlt300)=09842P(nlt150)=08488

454(a)no=285v=39(anormaldistributionisapplicable)A=98B=00202h1=ndash4814h2=5565h3=01452M[n |p0]=1021M[n |p1] =1010M[n]max=2194(b)n0=65v=8A=8B=02222h1ndash1861h2=2565h3=01452M[n|p0]=216M[n|p1]=386M[n]max=386

455 Apply the passage from a Poisson distribution to a chi-squaredistributionv=9n0=180A=18B=01053h1=ndash2178h2=2796h3=005123M[n|p0]=9086M[n|p1]=7982M[n]max=1252Foragoodlotifp=0wehavenmin=43foradefectivelotifp=1nmin=3

456

wherezparethequantilesofthenormaldistributionF(zp)=05+05Ψ(zp)=pz097=1881z092=1405z095=1645z090=1282z0=1613n0=87Thesinglesamplesizein thecaseofmagnitudecontrolfor thesameαβp0p1 isconsiderablysmallerthaninthecaseofcontroloftheproportionofdefectives

457 In thecaseofabinomialdistribution law(withpassage to thenormaldistributionlaw)α=01403β=01776n0=49v=6A=5864B=02065h1=ndash1945h2=2182h3=01452M[n|p0]=303M[n|p1]=264M[n]max= 342 The average expenditure in the case of double sampling for 100 lotsrepresents643430+356660=4070itemsInthecaseofsinglesamplingtheexpenditure of items for 100 lots is 4900 items in the case of sequentialanalysis the average expenditure for 100 good lots is not greater than 3030itemsInthecaseofaPoissondistributionα=01505β=02176n0=49v=6(passagetoachi-squaredistribution)

458Applythenormaldistributionlawn0=286v=15A=9900B=001h1=3529h2=7052h=004005M[n |002]=1760M[n |007]=2319M[n]max=6471c=3608P(nltM[n |002])=05993P(nltM[n |002])=09476P(nltn0)=08860

459Forn0=925v=12Fort0=1000hoursA=ndash2197B=2197t1=2376 t2 = ndash2376 t3 = 7499M[T | 10ndash5] = 6132M[T | 2middot10ndash5] = 4829M[T]max=7506

TABLE138

4510ForthemethodofsinglesamplingapplythepassagefromaPoissondistributiontoachi-squaredistributionv=6no=122A=184B=ndash008041h1=ndash1487h2=3077h3=00503Foragoodlotifp=0nmin=30foradefectivelotifp=1nmin=4

4511Foradoublesampleα=0001486β=00009152forasinglesamplen0 = 62 v = 13 (the passage to the normal distribution law)A = 6710B =00009166h1=ndash4446h2=4043h3=02485M[n|a0]=292M[n |a1]=160M[n]max=707Theaverageexpenditureofpotatoesper100 lots in thecaseofdoublesamplingis628840+371260=4743itemsTheexpenditureof potatoes per 100 lots is 6200 items In the case of sequential analysis theaverageexpenditureper100goodlotsisnotgreaterthan2920items

4512Foradoublesampleα=00896β=00233forasinglesamplen0=15v=1245A=10905B=002560h1=ndash9777h2=6372h3=1849M[n|σ0]=981M[n|σ1]=278M[n]max=10Inthecaseofdoublesamplingtheaverage expenditure of resistors per 100 good lots is 856613 + 144426 =1488inthecaseofsinglesamplingtheexpenditureis1500itemsinthecaseofsequentialanalysistheaverageexpenditureisnotlargerthan981items

4513 In the case of single sampling α = 00000884 β = 000621 B =000621A=112410h1=6506h2=ndash1194h3=5184M[n |ξ0]=2602M[n|ξ1]=4732M[n]max=1214c=2542P(nle300)gt099(lt0999)P(nle150)=09182

4514n0=86v=667hoursA=999B=0001001h1=6908h2 = ndash6908h3=6933λ=001442M[n|λ0]=2248M[n|λ1]=3567M[n]max=9931

4515Forasinglecontrolofproportionofunreliablecondensersn0=246v=5ForasequentialreliabilitycontrolofcondensersA=9999B=00001h1=1152104h2=ndash1152104h3=6384102λ=0000001566

4516tT=9526hoursv=728hourslnA=2197lnB=ndash2197

For thepoorerof thegood lots hours forthebetterofthedefectivelots hours


461Oneshouldprovethatif then

462Nosince butandconsequentlydoesnottendtozeroasTincreases

463

464

465466

467σywilldecreaseby2percent

468τywilldecreaseby3percent

4694610Thevalueofthefirstzeroofthefunction equals(a)220sec

(b)230sec4611

andthecorrespondingstandarddeviationsare241232219and171grad24612 When t increases the quotient t1t converges in probability to the

probability P of coincidence of the signs of the ordinates of the randomfunctions X(t) and X(t + τ) related for a normal process to the normalizedcorrelation function k(τ) by k(τ) = cos π(l ndash P) which can be proved byintegrating the two-dimensionalnormaldistribution lawof theordinatesof therandomfunctionbetweenproperlimits

4613Denotingby

andbyprobabilitythatsignsofX(t)andX(t+τ)coincideweget

Consequently

f(x1x2x3x4)beingthedistributionlawofthesystemofnormalvariablesX(t1)X(t1+τ)X(t2)X(t2+τ)

4614 wherewehave theapproximatereaction

ForTjexceedingconsiderablythedampingtimeofKx(τ)itisapproximatelytruethat

where and isasamplefunction4615

4616By9percent4617

4618Since

then

Themeanerroris =058middotA

1 For solution see Yaglom A M and Yaglon I M Challenging Mathematical Problems withElementarySolutionsSanFranciscoHolden-DayInc1964Problem92p29andsolutiontoproblem92pp202ndash209

2SeejahnkeEandEmdeRTableofFunctionswithFormulaeandCurves4threvedNewyorkDoverPublicationsInc1945

2SeeTablesofWeberParabolicCylinderFunctionsinFletcherAetalAnIndexofMathematicalTablesVolIIOxfordEnglandBlackwellScientificPublicationsLtd1962


1T The binomial coefficients Beyer W pp 339ndash340MiddletonD1960KoudenD1961pp564ndash567VolodinBGetal1962p393

2T ThefactorialsnorlogarithmsoffactorialslognBarlowP1962BeyerWpp449ndash450BronsteinIandSemendyaevKA1964BoevG1956pp350ndash353KoudenD1961pp 568ndash569 Segal B I and Semendyaev K A 1962 p393 Unkovskii V A 1953 p 311 Volodin B G et al1962p394

3T PowersofintegersBeyerWpp452ndash4534T ThebinomialdistributionfunctionP(dltm+1)=P(dlem)=

pk(1ndashp)nndashkBeyerWpp163ndash173KoudenD1961pp573ndash578

5T The values of the gamma-function Г(x) or logarithms of thegamma-function Г Beyer W p 497 Bronstein I andSemendyaev K A 1964 Hald A 1952 Middleton D1960BoevG1956p353SegalBIandSemendyaevKA1962pp353ndash391ShorYa1962p528

6TThe probabilities for a PoissondistributionBeyerWpp175ndash187GnedenkoBVSaatyT 1957 Boev G 1956 pp 357ndash358 Dunin-Barkovskii IV andSmirnovNV 1955 pp 492ndash494 SegalB I andSemendyaevKA1962

7T The totalprobabilities foraPoissondistributionBeyerWpp175ndash187

8T The Laplace function (the probability integral) in case of anargument expressed in terms of standard deviation

ArleyN andBuchK1950BeyerWpp115ndash124CrameacuterH1946GnedenkoBVandKhinchinA1962MilneWE1949PugachevVS1965SaatyT1957BernsteinS1946pp410ndash411

9T The probability density of the normal distribution

foranargumentexpressedinstandarddeviationsBeyerWpp115ndash124GnedenkoBVp383

10T The derivatives of the probability density of the normaldistributionφ(x)φ2(x)=φPrime(x)=(x2ndash1)φ(x)φ3(x)=φprimePrime(x)=ndash(x3ndash3x)φ(x)BeyerWpp115ndash124

11T The reduced Laplace function for an argument expressed in

standard deviations see8T

12T The probability density of the normal distribution for anargument expressed in standard deviation

see9T13T The function

see8T9T14T

TheStudentdistributionlaw

Beyer W pp 225ndash226Gnedenko B V YaglomA M and Yaglom I M1964VolodinBG et al1962 p 404 Segal B Iand Semendyaev K A196215T The probabilities

fortheStudentdistributionlawsee14T16T The values of γ associatedwith the confidence

level and k degrees offreedom for the Student distributionArleyNandBuchK1950CrameacuterH1946LaningJHJrandBattinRH1956UnkovskiiVA1953pp306ndash307seealso14T

17T The probabilities

forachi-squaredistributionon andkdegreesoffreedomBeyerWpp233ndash239GnedenkoBVMilneWE1949Dunin-Barkovskii IVandSmirnovNV1955pp505ndash507

18T The values of depending on the probabilityandkdegreesoffreedomforachi-

squaredistributionsee17T19T The lower limitγ1and theupper limitγ2of the

confidencelevelαandkdegreesoffreedomfora chi-square distributionLaning JH Jr andBattinRH1956SmirnovNVandDunin-BarkovskiiIV1959p405


forachi-squaredistributionsee22T21T The probability density of a chi-square

distribution

see5T9T22T The probabilities for the

quantity y obeying a chi-square distribution

BeyerWpp233ndash239ShorYa196223T The Rayleigh distribution law

Bartlett M1953

24T The function BartlettM1953

25T Theprobabilities

for theKolmogorovdistribution lawArleyNandBuchK1950GnedenkoBVMilneWE1949Dunin-Barkovskii IVandSmirnovNV1955pp539ndash540

26T The values of y(p-quantiles) depending on theparametercand theWalddistributionfunction

TakacsL1962BasharinovAandFleishmanB1962pp338ndash34427T

27T TablesofrandomnumbersBeyerWpp341ndash

34528T Thefunctionη(p)=ndashplog2pWaldA194729T

The orthogonalChebyshevpolynomials

MiddletonD196030T Two-sided confidence limits for the estimated

parameter in the binomial distribution law BeyerW187ndash189

31TThe values ofDwightH1958

32T The relations between the parametersbmvm andmfor theWeibulldistribution lawKoshlyakovNSGlinerEBandSmirnovMM1964

More complete information on the references is found in the Bibliography which follows thissection

BIBLIOGRAPHY

ArleyNandBuchKIntroductiontoProbabilityandStatisticsNewYorkJohnWileyandSonsInc1950

BachelierLCalculdesProbabiliteacutes(CalculusofProbabilities)Paris1942BarlowPBarlowrsquosTablesofSquaresCubesSquareRootsCubeRootsand

ReciprocalsofallIntegerNumbersupto125004thEdNewYorkChemicalPublishingCoInc1962

BartlettMPhilosophicalMagazineNo441953BasharinovAandFleishmanBMetodystatisticheskogoposledovateVnogo

analizaiikhprilosheniya(Methodsofstatisticalsequentialanalysisandtheirapplications)SovetskoeRadio1962

BernsteinSTeoriyaVeroyatnostei(ProbabilityTheory)Gostekhizdat1946BertrandICalculdesProbabiliteacutes(CalculusofProbabilities)Paris1897BeyerWHandbookofTablesforProbabilityandStatisticsChemicalRubber

CoOhioBoevGTeoriyaVeroyatnostei(ProbabilityTheory)Gostekhizdat1956BorelEElementsdelaTheacuteoriedesProbabiliteacutes(ElementsofProbability

Theory)Paris1924BronsteinIandSemendyaevKAGuideBooktoMathematicsfor

TechnologistsandEngineersNewYorkPergamonPressInc1964BunimovichVFluktuatsionnyeprotsessyvradio-priemnykhustroistvakh

(Randomprocessesinradio-receptionequipment)SovetskoeRadio1951CrameacuterHMathematicalMethodsofStatisticsPrincetonNJPrinceton

UniversityPress1946CzuberEWahrscheinlichkeitsrechnungundihreAnwendungauf

FehlerausgleichungStatistikundLebensversicherung(ProbabilityTheoryanditsApplicationtoError-SmoothingStatisticsandLifeInsurance)LeipzigandBerlin1910

DavenportWBJrandRootVLIntroductiontoRandomSignalsandNoiseNewYorkMcGraw-HillBookCoInc1958

DlinAMatematicheskayastatistikavtekhnike(Mathematicalstatisticsintechnology)SovetskayaNauka1958

Dunin-BarkovskiiIYandSmirnovNVTeoriyaVeroyatnosteiiMatematicheskayaStatistikavTekhnikemdashObshchayaChast(ProbabilityTheoryandMathematicalStatisticsinTechnologymdashGeneralPart)Gostekhizdat1955

DwightHMathematicalTablesofElementaryandSomeHigherOrderMathematicalFunctions3rdRevEdNewYorkDoverPublicationsInc1961

FellerWIntroductiontoProbabilityTheoryanditsApplicationsNewYorkJohnWileyandSonsIncVol11957Vol21966

GantmakherFRTheTheoryofMatricesNewYorkChelseaPublishingCo1959

GlivenkoVKursTeoriiVeroyatnostei(CourseinProbabilityTheory)GONTI1939

GnedenkoBVTheoryofProbabilityNewYorkChelseaPublishingCo(4thEdinprep)

GnedenkoBVandKhinchinAElementaryIntroductiontotheTheoryofProbability5thEdNewYorkDoverPublicationsInc1962

GoldmanSInformationTheoryEnglewoodCliffsNJPrentice-HallInc1953

GoncharovVTeoriyaVeroyatnostei(ProbabilityTheory)Oborongiz1939GuterRSandOvchinskiiBVElementyChislennogoAnalizai

MatematicheskoiObrabotkiResuVtatovOpita(ElementsofNumeralAnalysisandtheMathematicalProcessingofExperimentalData)Fizmatgiz1962

GyunterNMandKuzrsquominROSbornikZadachpoVyssheiMatematikemdashChIII(CollectionofProblemsinHigherMathematicsmdashPartIII)Gostekhizdat1951

HaldAStatisticalTheorywithEngineeringApplicationsNewYorkJohnWileyandSonsInc1952

JahnkeEandEmdeFTablesofFunctionswithFormulaeandCurvesNewYorkDoverPublicationsInc1945

KadyrovMTablitsySluchainykhChisel(TableofRandomNumbers)Tashkent1936

KhinchinARabotypoMatematicheskoiTeoriiMassovogoObsluzjevaniya(WorkintheMathematicalTheoryofMassService[Queues])Fizmatgiz1963

KoshlyakovNSGlinerEBandSmirnovMMDifferentialEquationsofMathematicalPhysicsNewYorkJohnWileyandSonsInc(Interscience)1964

KotelrsquonikovVAnomogramconnectingtheparametersofWeibullrsquosdistributionwithprobabilitiesTheoryofProbabilityandItsApplications9670ndash6741964

KoudenDStatischeskieMetodyKontrolyaKachestva(StatisticalMethodsof

QualityControl)Fizmatgiz1961KrylovVIApproximateCalculationsofIntegralsNewYorkTheMacmillan

Co1962LaningJHJrandBattinRHRandomProcessesinAutomaticControl

NewYorkMcGraw-HillBookCoInc1956LevinBTeoriyasluchainykhprotsessovieeprimenenievradiotekhnike

(Theoryofrandomprocessesanditsapplicationtoradiotechnology)SovetskoeRadio1957

LinnikYYMethodofLeastSquaresandPrinciplesoftheTheoryofObservationsNewYorkPergamonPressInc1961

LukomskiiYaTeoriyaKorrelyatsiiieePrimeneniekAnalizuProizvodstva(CorrelationTheoryanditsApplicationtotheAnalysisofProduction)Gostekhizdat1961

MesyatsevPPPrimenenieTeoriiVeroyatnosteiiMatematicheskoiStatistikipriKonstruirovanniiiProizvodstveRadio-Apparatury(ApplicationsofProbabilityTheoryandMathematicalStatisticstotheConstructionandProductionofRadios)Voenizdat1958

MiddletonDIntroductiontoStatisticalCommunicationTheoryNewYorkMcGraw-HillBookCoInc1960

MilneWENumericalCalculusPrincetonNJPrincetonUniversityPress1949

NalimovVVApplicationofMathematicalStatisticstoChemicalAnalysisReadingMassAddison-WesleyPublishingCoInc1963

PugachevVSTheoryofRandomFunctionsReadingMassAddison-WesleyPublishingCoInc1965

RomanovskiiVDiskretnyeTsepiMarkova(DiscreteMarkovChains)Gostekhizdat1949

RomanovskiiVMatematicheskayaStatistika(MathematicalStatistics)GONTI1938

RumshiskiiLZElementsofProbabilityTheoryNewYorkPergamonPressInc1965

SaatyTResumeacuteofusefulformulasinqueuingtheoryOperationsResearchNo21957

SarymsakovTAOsnovyTeoriiProtsessovMarkova(BasicTheoryofMarkovProcesses)Gostekhizdat1954

SegalBIandSemendyaevKAPyatiznachnyeMatematicheskieTablitsy(Five-PlaceMathematicalTables)Fizmatgiz1961

ShchigolevBMMathematicalAnalysisofObservationsNewYorkAmericanElsevierPublishingCoInc1965

SherstobitovVVandDinerISbornikZadachpoStreVbezenitoiArtilrii(CollectionofProblemsinAntiaircraftArtilleryFirinz)Voenizdat1948

ShorYaStatisticheskiemetodyanalizaikontrolyakachestvainadezhnosti(Statisticalmethodsofanalysisqualitycontrolandsafety)SovetskoeRadio1962

SmirnovNVandDunin-BarkovskiiIVKratkiiKursMatematicheskoiStatistiki(ShortCourseinMathematicalStatistics)Fizmatgiz1959

SolodovnikovVStatisticalDynamicsofLinearAutomaticControlSystemsPrincetonNJDVanNostrandCoInc1956

StratonovichRLIzbrannyevoprosyteoriifluktuatsiivradioteknike(Selectedquestionsinfluctuationtheoryinradiotechnology)SovetskoeRadio1961

SveshnikovAAAppliedMethodsoftheTheoryofRandomFunctionsNewYorkPergamonPressInc(inprep)

TakacsLStochasticProcessesProblemsandSolutionsNewYorkJohnWileyandSonsInc1960

UnkovskiiVATeoriyaVeroyatnostei(ProbabilityTheory)Voenmorizdat1953

UorsingAandGeffnerDMetodyObrabotkiEksperimentaVnykhDannykh(MethodsforProcessingExperimentalData)IL1953

VenttselrsquoESTeoriyaveroyatnostei(Probabilitytheory)IzdrsquovoNauka1964VolodinBGetalRukovodstvoDlyaInzhenerovpoResheniyuZadachTeorii

Veroyatnostey(EngineerrsquosGuidefortheSolutionofProblemsinProbabilityTheory)Sudpromgiz1962

WaldASequentialAnalysisNewYorkJohnWileyandSonsInc1947YaglomAMandYaglomIMChallengingMathematicalProblemswith

ElementarySolutionsSanFranciscoHolden-DayInc1964YaglomAMandYaglomIMProbabilityandInformationNewYork

DoverPublicationsInc1962YuleGUandKendallMGIntroductoryTheoryofStatistics14thRevEd

NewYorkHafnerPublishingCoInc1958

Index

Absorbingstate232

Additionofprobabilities16ndash22AftereffectandMarkovprocess248Apolloniusrsquotheorem147Arctanlaw321Arithmeticmeandeviation73Asymmetrycoefficient108

Bayesrsquoformula26ndash30Besselformulas329Binomialdistribution30

Cauchydistribution321Cauchyprobabilitylaw53120Centralmomentcomputationof62definitionof54

Characteristicfunction74ndash79ofrandomvariables108subsystemsof125systemsof124ndash128

Charlier-Aseries302Chebyshevrsquosinequality171Chebyshevrsquospolynomials327Chebyshevrsquostheorem171Chi-squaretest301Complementaryevents1Compositionofdistributionlaws128ndash136Conditionaldifferentialentropy157Conditionaldistributionlaws99ndash106

Conditionalentropy157Conditionalmeanentropy158Conditionalprobability12ndash16Conditionalvariance103Confidenceintervals286ndash300Confidencelevels286ndash300ContinuousMarkovprocesses256ndash274Continuousrandomvariables48ndash53numericalcharacteristicsof62ndash67

Convolutionofdistributionlaws128ndash136Correlationcoefficient85Correlationtheoryofrandomfunctions181ndash230propertiesof181ndash185

Covarianceofrandomvariables85

Dcomputationof62definitionof548-function49

Dataprocessingmethodsof275ndash374Degeneratenormaldistribution145DeMoivre-Laplacetheorem176ndash180Dependentevents12Deviationvectorsuseof145ndash156Differentialentropy157Differentialequations205Discreterandomvariable43ndash48numericalcharacteristicsof54ndash62

Distributionellipse146Distributionfunction43ndash48Distributionlaws84ndash91compositionof128ndash136convolutionof128ndash136offunctionsofrandomvariables115ndash123ofrandomfunctions181ndash185symmetric62

Distributionpolygon43ndash48Doublesampling348Dynamicalsystemscharacteristicsatoutputof205ndash216

EncodingShannon-Fanomethod163Entropyandinformation157ndash170ofrandomeventsandvariables157ndash162

Envelopesmethodof226ndash230Erlangrsquosformula253Essentialstates232Estimatesofrandomvariables275Excessofrandomvariable108Expectationcomputationof62definitionof54

Exponentialdistribution319

Fokker-Planckequation256

Generatingfunction36ndash42Geometricprobability6ndash11Goodness-of-fittestsof300ndash325Greenrsquosfunction206

HomogeneousMarkovchain231HomogeneousMarkovprocess297Homogeneousoperator185Hypergeometricdistribution313

Impulsefunction206Independentevents12Independenttrialsrepeated30ndash36Informationandentropy157ndash170quantityof163ndash170

Integraldistributionlaw43Intersectionofevents1IrreducibleMarkovchain231

Jacobiandeterminant116

Khinchinrsquostheorem171Kolmogorovequations256Kolmogorovtest301

Lagrange-Sylvesterformula231Laplacefunction71normalized71

Largenumberslawof171ndash175Leastsquaresdataprocessingby325ndash346Limittheorems171ndash180Linearizationoffunctionsofrandomvariables136ndash145Linearoperationswithrandomfunctions185ndash192Linearoperator185Logarithmicnormaldistributionlaw53Lyapunovtheorem176ndash180

Mcomputationof62definitionof54

mkcomputationof62definitionof54

microkcomputationof62definitionof54

Markovchains231ndash246Markovprocesses231ndash274withdiscretenumberofstates246ndash256

Markovrsquostheorem171Maximaldifferentialentropy159Maxwelldistribution319Meandeviation62arithmetic73

Meanerror72Mean-squaredeviationcomputationof62definitionof54

Median49Mode49Moment(s)centralcomputationof62definitionof54

computationof62definitionof54ofrandomvariables275ndash286

Multidimensionalnormaldistribution91ndash99MultidimensionalPoissonlaw70Multinomialdistribution36ndash4270Multiplicationofprobabilities12ndash16Mutualcorrelationfunction182Mutuallyexclusiveevents1

Nonhomogeneousoperator185Normaldistributionlaw70ndash7491ndash99Normalizedcovariancematrix85NormalizedLaplacefunction71

Optimaldynamicalsystems216ndash225OrdinarityofMarkovprocess248

Pascalrsquosdistributionlaw78Passagesproblemson192ndash198Pearsonrsquoslaw120Pearsonrsquostests302PeriodicMarkovchain231Perronformula232Poissonrsquoslaw67ndash70Probability(ies)additionof12ndash16characteristicsofdeterminationof368ndash374conditional12ndash16evaluationofdirectmethodfor4ndash6geometric6ndash11multiplicationof12ndash16total22ndash26

Probabilitydensitycomputationof80ndash83Probabilitydensityfunction48ndash53Probabilitydistributionseries43ndash48Probabilityintegral71

Qualitycontroldefinitionof346statisticalmethodsfor346ndash368

Quantile49

Randomevent(s)1ndash42relationsamong1ndash3

Randomfunction(s)correlationtheoryof181ndash230definitionof181distributionlawsof181ndash185linearoperationswith185ndash192stationary181

Randomsequence181Randomvariable(s)43ndash83continuous48ndash53numericalcharacteristicsof62ndash67discrete43ndash48numericalcharacteristicsof54ndash62excessof108functionsof107ndash157distributionlawsof115ndash123linearizationof136ndash145numericalcharacteristicsof107ndash115momentsof275ndash286systemsof84ndash106characteristicsof84ndash91uncorrelated85

Rayleighdistribution52318Rayleighrsquoslaw119Recursionformulas36ndash42RegularMarkovprocess247Repeatedindependenttrials30ndash36acomputationof62definitionof54

Sequentialanalysis349Setofexperimentscomplete1Shannon-Fanomethodofencoding163

Sheppardcorrections277Simpsondistribution315Singlesampling346Spectraldecompositionofstationaryrandomfunctions198ndash205Spectraldensity198Standarddeviation62Stateabsorbing232essential232

StationarityofMarkovprocess248Stationaryrandomfunction181spectraldecompositionof198ndash205

Stochasticprocess181Studentrsquosdistribution287Symmetricdistributionlaw62

Totalprobability22ndash26computationof80ndash83

Transitionprobability231TransitiveMarkovprocess248Transmissionfunction217Triangulardistribution315

Unbiasedestimateofrandomvariables275Uniformdistribution52Unionofevents1

Variancecomputationof62definitionof54

Waldanalysis349Weibulldistributionfunction52319

Title Page

Copyright Page

Foreward

Contents

I Random Events

1 Relations Among Random Events

2 A Direct Method for Evaluating Probabilities

3 Geometric Probabilities

4 Conditional Probability The Multiplication Theorem for Probabilities

5 The Addition Theorem for Probabilities

6 The Total Probability Formula

7 Computation of The Probabilities of Hypotheses after A Trial (Bayesrsquo Formula)

8 Evaluation of Probabilities of Occurrence of An Event in Repeated Independent Trials

9 The Multinomial Distribution Recursion Formulas Generating Functions

II Random Variables

10 The Probability Distribution Series The Distribution Polygon and The Distribution Function of A Discrete Random Variable

11 The Distribution Function and The Probability Density Function of A Continuous Random Variable

12 Numerical Characteristics of Discrete Random Variables

13 Numerical Characteristics of Continuous Random Variables

14 Poissonrsquos Law

15 The Normal Distribution Law

16 Characteristic Functions

17 The Computation of The Total Probability and The Probability Density in Terms of Conditional Probability

III Systems of Random Variables

18 Distribution Laws and Numerical Characteristics of Systems of Random Variables

19 The Normal Distribution Law in The Plane and in Space The Multidimensional Normal Distribution

20 Distribution Laws of Subsystems of Continuous Random Variables and Conditional Distribution Laws

IV Numerical Characteristics and Distribution Laws of Functions of Random Variables

21 Numerical Characteristics of Functions of Random Variables

22 The Distribution Laws of Functions of Random Variables

23 The Characteristic Functions of Systems and Functions of Random Variables

24 Convolution of Distribution Laws

25 The Linearization of Functions of Random Variables

26 The Convolution of Two-Dimensional and Three-Dimensional Normal Distribution Laws by use of The Notion of Deviation Vectors

V Entropy and Information

27 The Entropy of Random Events and Variables

28 The Quantity of Information

VI The Limit Theorems

29 The Law of Large Numbers

30 The de Moivre-Laplace and Lyapunov Theorems

VII The Correlation Theory of Random Functions

31 General Properties of Correlation Functions and Distribution Laws of Random Functions

32 Linear Operations with Random Functions

33 Problems on Passages

34 Spectral Decomposition of Stationary Random Functions

35 Computation of Probability Characteristics of Random Functions at The Output of Dynamical Systems

36 Optimal Dynamical Systems

37 The Method of Envelopes

VIII Markov Processes

38 Markov Chains

39 The Markov Processes with A Discrete Number of States

40 Continuous Markov Processes

IX Methods of Data Processing

41 Determination of The Moments of Random Variables from Experimental Data

42 Confidence Levels and Confidence Intervals

43 Tests of Goodness-of-Fit

44 Data Processing by The Method of Least Squares

45 Statistical Methods of Quality Control

46 Determination of Probability Characteristics of Random Functions from Experimental Data

Answers and Solutions

Sources of Tables Referred to in The Text

Bibliography

Index


EditedbyAASVESHNIKOV

TranslatedbyScriptaTechnicaIncEditedbyBernardRGelbaum

DOVERPUBLICATIONSINCNEWYORK


Conventions







BernardRGelbaum











II RANDOMVARIABLES





14 Poissonrsquoslaw



















VI THELIMITTHEOREMS












38 Markovchains










ANSWERSANDSOLUTIONS


BIBLIOGRAPHY

INDEX

I RANDOMEVENTS





















PROBLEMS




FIGURE1














PROBABILITIES

BasicFormulas














andhence


PROBLEMS







































FIGURE2

FIGURE3





FIGURE4

FIGURE5


andtherefore





PROBLEMS















































PROBLEMS















































Analogously Then




TABLE1



Since



FIGURE6




Fromthiswefind










andfinally


or


PROBLEMS
































530Provethat












where




where




Wehave










PROBLEMS

























givenbytheformula

where















PROBLEMS










































FIGURE7












PROBLEMS













FIGURE8


























































thatis















Wehave



PROBLEMS
































II RANDOMVARIABLES











TABLE2

FIGURE9







TABLE3






TABLE4












FIGURE10








FIGURE11



PROBLEMS























(3)

(4)


(1)f(x)ge0

(2)

(3)

(4)












weobtain





F(x05)=12

Solvingtheequation





Thisimpliesthat








TABLE5



PROBLEMS




























BasicFormulas





orby











Consequentlywehave





Consequently





















whosesumgives

or



TABLE6



ForthedeviceB




PROBLEMS
















(a)forthedeviceA

(b)forthedeviceB

FIGURE12






































whichforevennare

where

andforoddn



Thus

(b)Since

then

(c)

whereConsequently






PROBLEMS











(Maxwellrsquoslaw)
























density



expectation

14 POISSONrsquoSLAW















FIGURE13


PROBLEMS
















inwhich

and





form

or



inwhich


inwhich








Fromthisweget


andfinally











PROBLEMS
























BasicFormulas
















thisleadsto

thatis









FIGURE14


Similarlyforxgt0












andthemoments


PROBLEMS


































weobtain


Consequently

Since

then



and





or

PROBLEMS
































thevariances

andthecovariances









independentif








Similarlywefindthat






ThevarianceofXis


FIGURE15

Thecovarianceis








PROBLEMS







TABLE7






FIGURE16








1815Showthat


(X1X2X3)is








BasicFormulas









where















SOLUTION(a)Since

where

then

Thisimpliesthat








inwhich



PROBLEMS




the plane


















FIGURE17









FIGURE18
























SOLUTION(a)Since




Similarly

and




weobtain


since






weobtain

or

Similarlywefindthat


andsimilarly


Similarly





Consequently

FIGURE19



PROBLEMS






















distributionlaw
















then








where




Therefore




Therefore





Then


where

Finally






FIGURE20




PROBLEMS





















































FIGURE21








where

FIGURE22

Finallywehave





and





FIGURE23





for0ltzlt14

forndash14ltzlt0

FIGURE24A

FIGURE24B







Therefore




PROBLEMS



























distributionlaw




























wehave




equals






and

Therefore


or








Then

whichimpliesthat


PROBLEMS

































where




FIGURE25





FIGURE26

FIGURE27






FIGURE28

















Consequently



PROBLEMS
















where





























where


TABLE8



where





wherekvθ=rσvσθ





where



where



PROBLEMS

































FIGURE29







FIGURE30
















givenbytheformulas



FIGURE31




where


TABLE9






where


where






andconsequently







thatis


TABLE10



TABLE11

Check







Thus


PROBLEMS




TABLE12


TABLE13








aregiveninTable14

TABLE14









FIGURE32








TABLE15



FIGURE33





















and








TABLE17


andforthesecondone




underconstraints



undertheconstraints

and



andtherfore

where



distributionlaw




Itfollowsthat


PROBLEMS







































TABLE18







Thisimpliesthat

















TABLE19







PROBLEMS


























precedingletters

TABLE20











VI THELIMITTHEOREMS


BasicFormulas














inequalities














aπ then

Since







Wehave



PROBLEMS









TABLE21



TABLE22











BasicFormulas




















obtain

or














PROBLEMS

































distributionlaws

















then






PROBLEMS

311Provethat



















isgivenintheform




















andforthevariance

Since









obtains


Therequiredvariance








PROBLEMS













X(t)


























3225Given














known







Fornormalfunctions















Since

wehave

















Then





weobtain

thatis



PROBLEMS







































differentiableis




















weseethat

Since




SOLUTIONSince

PROBLEMS















FindSΔ(ω)if





















FIGURE34












wehave






Inthiscase




















Wehave

where

Since

then

and



if




PROBLEMS
















358Doestheequation














wherekgehgt0


3514Given



















3523Giventhesystem






























U(t)then

where




where








where















SOLUTIONInthiscase











PROBLEMS

















































if



















where


Similarly









randomfunction



Consequently




where


PROBLEMS







if



















38 MARKOVCHAINS





















































andconsequently












Then









Weset


Fromthiswefindthat





















whichcanbewrittenas


Thisimpliesthat












PROBLEMS












where






































relation








where

















applicable



Sincebydefinition

wehave










fornge1and













whichleadsto


















PROBLEMS

































where





































thatis









Consequently



equation


where















Henceitfollowsthat










where















Since


















wefindthatthatis









where







Thus



weobtainthat

PROBLEMS










where



wherethepolynomials





equations

where













andconsideringthat


where

























BasicFormulas






where

TABLE23






TABLE24







TABLE25







TABLE26





TABLE27





Therefore

where(jnei)


Here

Finallyweget


TABLE28








TABLE29


TABLE30




PROBLEMS






TABLE31







TABLE32





where












TABLE33

TABLE34


where



TABLE35



TABLE36




BasicFormulas






forunknownσ

where




where





Fromthis















and


Ifm=0thenp1=0and

Ifm=nthenp2=1and



(the













Therandomvariable


Then

Solvingtheequation







hours

















TABLE37


TABLE38






FIGURE35








Fromthis















Inthiscase

and


088(10141768)for 12040(00530808)for 13


(077094)for 12(005067)for 13

PROBLEMS



TABLE40




TABLE41






TABLE42

























TABLE44





BasicFormulas




















respectively
















TABLE46




TABLE47





TABLE48





Wefindthat






TABLE49






TABLE50


Computing








TABLE51



where

wefind


TABLE52










TABLE53



TABLE54


Determining




TABLE55




andY


TABLE56



TABLE57






PROBLEMS



TABLE58


TABLE59




TABLE60


TABLE61



TABLE62








TABLE63



TABLE64






TABLE65





TABLE66



TABLE67


TABLE68



TABLE69






TABLE70





TABLE71





TABLE72




TABLE73







where



where



TABLE75


FIGURE36

TABLE76





TABLE77



TABLE78






TABLE79



where



TABLE80



TABLE81







TABLE82



BasicFormulas





where































TABLE83



where












TABLE84

If





TABLE85

Weobtain






TABLE86



where




become





TABLE87




TABLE88








TABLE89

Weobtain


wefind



TABLE90





TABLE91









become


or


TABLE92




TABLE93

Weobtain




andget


TABLE94


andobtain


givenbytheformulas

Wehave


orby




TABLE95


TABLE96




TABLE97

weobtion









PROBLEMS


TABLE98







TABLE99



TABLE100





TABLE101




TABLE102






TABLE103



TABLE104




TABLE105




stream

TABLE106



TABLE107




TABLE108

If



where


TABLE109




TABLE110

Iftherelation



BasicFormulas











where







then



then







where



where







where



Ifnle01Np1lt01then



FIGURE37













where












where
















andobtain








Wehave
























distributionweget




FIGURE38





fork=16




TABLE111


t2=ndash5353hours



Furthermorewefind

FIGURE39

thenwecompute



Forγ=05wehave




TABLE112



(Figure40)are

Wecomputenmin





FIGURE40
























or since λ0 = 1t0 = 000078 λ1 = 1t1 =








PROBLEMS








































formulasbecome

or




















PROBLEMS










form







































where



ANSWERSANDSOLUTIONS

I RANDOMEVENTS

















210

211

21221303

214

215


217

218



221




31

32



35














for


320321



Formlen2


Forn3lem2

Forn3gem2


weobtain


FIGURE41



weshallhave





49




419

420421

422

423

424

425P(A)=P(B)=P(C)=



428

429

430

431






510035205553pk= pkj542(rR)255112656







523


525p+q=1q= pp=

















wherebeingthe




61




Consider








615





71









marksmanfails











81(a)094=0656(b)094+4middot01middot093=0948

82 (a) (b)

83






TABLE114

815p=1ndash09510=04816p=1ndash095=041817



821(a) (b)3p4ndash4p3+ =0p=0614822

823 8240784825The200wones(R61=0394R102=0117)82606482702816828

829 830Werequire

831 We require

832P4 0=03024P4 1=04404P4 2=02144P4 3 = 00404P4 4 =00024

833026834015983595144



91p=P5221+2P5320=5024392p=P3111+P3210+P3120=0245

93(a) (b)

94 95

96 (a) (b)

97(a)





Thus







generatingfunction









Forn=10m=20









or


TABLE115



thatis

II RANDOMVARIABLES










TABLE120



111

112113

114

115(a)σ(b) (c)


p)1m(c)

117 (a) 10 (b) where

118 (a) (b) (c)

119


1111p=



1114

1115








129

1210

1211













1230


since


131M[X]=aD[X]= E=

132M[X]=0D[X]=

133

134 135

136 137M[X]=D[X]=m+1

138 139M[X]=0D[X]=2

1310 M[X]=(α+1)βD[X]=β2(α+1)1311

1312



1313

1314Usetherelation








1320

1321M[X]=0D[X]=

1322 wheremj=M[Xj]

1323 where

14 POISSONrsquoSLAW



144p=endash05asymp061145(1)095958(2)095963146091470143

148 14904

1410Sk=








155 156SeeTable122

TABLE122



1518





166

167E(u)= mk=k

168

169 wherev=u2hand


1610





1613







pointsbylweobtain

173p=015174


177

178

179

1710



181


182 (a) A = 20 (b)



TABLE123


(c)M[X]=1947M[Y]=0504

186





1812



18161817

1818


1820

1821



191192

193

194195

196

197


199



1912

1913

1914

1915

19161917

1918

wherep2=0196p3=0198p4=0148p5=0055q=0403

1919P= [Φ(k)]2

19201921

1922




201

202For|x|leR|y|leR



204205





2010



2011

2012

2013M[X|y]=08y+149M[Y|x]=045xndash8625

2014


2016

2017

2018

2019



2114aπ212π(a2)213M[G]=41gD[G]=032g2


2110211111a218π21123π2113

2114


2118


2119


2120

2121


21262127

2128(a)267sqm(b)220sqm(c)10sqm2129

21302131

2132M[Z]=0D[Z]=2Δ2σ2

21332134



223

224

225

226

227

228

229(b)ifagt0then

ifalt0then

(c)2210Foranoddn

forevenn

2211

22122213

2214

22162217

2218

2219


2220

2221







2226

2227




233234



238

239

231023112312




2317


241242

243

where

244



where


TABLE124


247

248249


2412


2415

2416

24172418SeeTable125

TABLE125




253

254





2516

2517





2521199mg

2522

2523







266(a)(a)a=b=25 m(b)a=689mb=388mα=15deg




FIGURE42





courseoftheship2617


2619


FIGURE43

FIGURE44


Fromthiswefind


where



271Since






278H[X]=loga(05 )


2710




27162717loga1050andloga302718











ForcodeNo2


288


(b)SeeTable127

TABLE127

(c)SeeTable128

TABLE128


(a)



TABLE129

TABLE130






where


2816

where

VI THELIMITTHEOREMS


291









298


(b)satisfiedsince





where




2913Consider











307nasymp65308p=0943309675






3015











Consequently










3110

3111


thatis

3113








=0P=030853211

3212

32133214



32173218D[Y(20)]=1360cm23219

3220





2τ2[1+2α2(1ndash2α2τ2)]

32253226




3231

where






Answer



338For ge545m

339







3313

where




where


341

342343Denoting

wehave

344345

346



34103411



3412Since

then

3413Since

then

3414Since


3415

34163417Since


where

3418Since then


where



34213422

3423



3424Since then


3425

where







353

where




355


356

where








3515



origin




3519

3520

3521

3522



and

then

3524









andforτle0




where

365

where

366

367

where

where


3611

where

3612

where

3613 3614

where


3616 where


where

3618

where






36223623

3624

3625

3626


371

where


372Since





378


then

3710Since




371300424αsecndash13714

where

3715


38 MARKOVCHAINS



Perron formula



where


ForigtkAki(λ)=0

Forkgti


ForigtkAki(λ)=0

Forkgtiknem







m) =0foriltkForigek






where



where






Theeigenvaluesare

where




Ifλ2=1then




weget



Then




itfollowsthat

Since






thatis


itfollowsthat

Theseries





thatis


weobtain







3827





qwehave



391

392 393

where


394 395





hasthesolutions







hasthesolutions












where

3915


hasthesolutions






where

thusitfollowsthat




401

402


whicha1=x2




408


409





where



where


where





where

40174018

where




4020

where






417

418

419

4110



4114 =4831m =5331m =1075m =1250m4115

where


4116 =23m =107m4117







421(9236m10764m)




TABLE135


4226

















4314 =875 =1685 =1186kH=5 =00398an




4316




4317 = 8746 = 2471 = 8002 = 9490 gt 500 kH = 7













4325


4326 =1153 =2143 =1020kH=10 =043


where



where




441 =0609+01242EM00=03896M11=000001156 =1464

=05704 =00000169442 =0679+0124E =1450 =05639 =000001672The


443


445

446




TABLE136

448 =03548+006574x+000130x2 =00147 =00106 =000156

449


TABLE137


44114412


4414

4415 =ndash3924+1306x|εmax|=141








456





TABLE138












463

464

465466




(b)230sec4611



4613Denotingby


Consequently





4616By9percent4617

4618Since

then






















see8T9T14T












distribution




Bartlett M1953














BIBLIOGRAPHY


































































Index

Absorbingstate232







































Quantile49














Title Page

Copyright Page

Foreward

Contents

I Random Events










II Random Variables



































38 Markov Chains












Bibliography

Index


Conventions







BernardRGelbaum











II RANDOMVARIABLES





14 Poissonrsquoslaw



















VI THELIMITTHEOREMS












38 Markovchains










ANSWERSANDSOLUTIONS


BIBLIOGRAPHY

INDEX

I RANDOMEVENTS





















PROBLEMS




FIGURE1














PROBABILITIES

BasicFormulas














andhence


PROBLEMS







































FIGURE2

FIGURE3





FIGURE4

FIGURE5


andtherefore





PROBLEMS















































PROBLEMS















































Analogously Then




TABLE1



Since



FIGURE6




Fromthiswefind










andfinally


or


PROBLEMS
































530Provethat












where




where




Wehave










PROBLEMS

























givenbytheformula

where















PROBLEMS










































FIGURE7












PROBLEMS













FIGURE8


























































thatis















Wehave



PROBLEMS
































II RANDOMVARIABLES











TABLE2

FIGURE9







TABLE3






TABLE4












FIGURE10








FIGURE11



PROBLEMS























(3)

(4)


(1)f(x)ge0

(2)

(3)

(4)












weobtain





F(x05)=12

Solvingtheequation





Thisimpliesthat








TABLE5



PROBLEMS




























BasicFormulas





orby











Consequentlywehave





Consequently





















whosesumgives

or



TABLE6



ForthedeviceB




PROBLEMS
















(a)forthedeviceA

(b)forthedeviceB

FIGURE12






































whichforevennare

where

andforoddn



Thus

(b)Since

then

(c)

whereConsequently






PROBLEMS











(Maxwellrsquoslaw)
























density



expectation

14 POISSONrsquoSLAW















FIGURE13


PROBLEMS
















inwhich

and





form

or



inwhich


inwhich








Fromthisweget


andfinally











PROBLEMS
























BasicFormulas
















thisleadsto

thatis









FIGURE14


Similarlyforxgt0












andthemoments


PROBLEMS


































weobtain


Consequently

Since

then



and





or

PROBLEMS
































thevariances

andthecovariances









independentif








Similarlywefindthat






ThevarianceofXis


FIGURE15

Thecovarianceis








PROBLEMS







TABLE7






FIGURE16








1815Showthat


(X1X2X3)is








BasicFormulas









where















SOLUTION(a)Since

where

then

Thisimpliesthat








inwhich



PROBLEMS




the plane


















FIGURE17









FIGURE18
























SOLUTION(a)Since




Similarly

and




weobtain


since






weobtain

or

Similarlywefindthat


andsimilarly


Similarly





Consequently

FIGURE19



PROBLEMS






















distributionlaw
















then








where




Therefore




Therefore





Then


where

Finally






FIGURE20




PROBLEMS





















































FIGURE21








where

FIGURE22

Finallywehave





and





FIGURE23





for0ltzlt14

forndash14ltzlt0

FIGURE24A

FIGURE24B







Therefore




PROBLEMS



























distributionlaw




























wehave




equals






and

Therefore


or








Then

whichimpliesthat


PROBLEMS

































where




FIGURE25





FIGURE26

FIGURE27






FIGURE28

















Consequently



PROBLEMS
















where





























where


TABLE8



where





wherekvθ=rσvσθ





where



where



PROBLEMS

































FIGURE29







FIGURE30
















givenbytheformulas



FIGURE31




where


TABLE9






where


where






andconsequently







thatis


TABLE10



TABLE11

Check







Thus


PROBLEMS




TABLE12


TABLE13








aregiveninTable14

TABLE14









FIGURE32








TABLE15



FIGURE33





















and








TABLE17


andforthesecondone




underconstraints



undertheconstraints

and



andtherfore

where



distributionlaw




Itfollowsthat


PROBLEMS







































TABLE18







Thisimpliesthat

















TABLE19







PROBLEMS


























precedingletters

TABLE20











VI THELIMITTHEOREMS


BasicFormulas














inequalities














aπ then

Since







Wehave



PROBLEMS









TABLE21



TABLE22











BasicFormulas




















obtain

or














PROBLEMS

































distributionlaws

















then






PROBLEMS

311Provethat



















isgivenintheform




















andforthevariance

Since









obtains


Therequiredvariance








PROBLEMS













X(t)


























3225Given














known







Fornormalfunctions















Since

wehave

















Then





weobtain

thatis



PROBLEMS







































differentiableis




















weseethat

Since




SOLUTIONSince

PROBLEMS















FindSΔ(ω)if





















FIGURE34












wehave






Inthiscase




















Wehave

where

Since

then

and



if




PROBLEMS
















358Doestheequation














wherekgehgt0


3514Given



















3523Giventhesystem






























U(t)then

where




where








where















SOLUTIONInthiscase











PROBLEMS

















































if



















where


Similarly









randomfunction



Consequently




where


PROBLEMS







if



















38 MARKOVCHAINS





















































andconsequently












Then









Weset


Fromthiswefindthat





















whichcanbewrittenas


Thisimpliesthat












PROBLEMS












where






































relation








where

















applicable



Sincebydefinition

wehave










fornge1and













whichleadsto


















PROBLEMS

































where





































thatis









Consequently



equation


where















Henceitfollowsthat










where















Since


















wefindthatthatis









where







Thus



weobtainthat

PROBLEMS










where



wherethepolynomials





equations

where













andconsideringthat


where

























BasicFormulas






where

TABLE23






TABLE24







TABLE25







TABLE26





TABLE27





Therefore

where(jnei)


Here

Finallyweget


TABLE28








TABLE29


TABLE30




PROBLEMS






TABLE31







TABLE32





where












TABLE33

TABLE34


where



TABLE35



TABLE36




BasicFormulas






forunknownσ

where




where





Fromthis















and


Ifm=0thenp1=0and

Ifm=nthenp2=1and



(the













Therandomvariable


Then

Solvingtheequation







hours

















TABLE37


TABLE38






FIGURE35








Fromthis















Inthiscase

and


088(10141768)for 12040(00530808)for 13


(077094)for 12(005067)for 13

PROBLEMS



TABLE40




TABLE41






TABLE42

























TABLE44





BasicFormulas




















respectively
















TABLE46




TABLE47





TABLE48





Wefindthat






TABLE49






TABLE50


Computing








TABLE51



where

wefind


TABLE52










TABLE53



TABLE54


Determining




TABLE55




andY


TABLE56



TABLE57






PROBLEMS



TABLE58


TABLE59




TABLE60


TABLE61



TABLE62








TABLE63



TABLE64






TABLE65





TABLE66



TABLE67


TABLE68



TABLE69






TABLE70





TABLE71





TABLE72




TABLE73







where



where



TABLE75


FIGURE36

TABLE76





TABLE77



TABLE78






TABLE79



where



TABLE80



TABLE81







TABLE82



BasicFormulas





where































TABLE83



where












TABLE84

If





TABLE85

Weobtain






TABLE86



where




become





TABLE87




TABLE88








TABLE89

Weobtain


wefind



TABLE90





TABLE91









become


or


TABLE92




TABLE93

Weobtain




andget


TABLE94


andobtain


givenbytheformulas

Wehave


orby




TABLE95


TABLE96




TABLE97

weobtion









PROBLEMS


TABLE98







TABLE99



TABLE100





TABLE101




TABLE102






TABLE103



TABLE104




TABLE105




stream

TABLE106



TABLE107




TABLE108

If



where


TABLE109




TABLE110

Iftherelation



BasicFormulas











where







then



then







where



where







where



Ifnle01Np1lt01then



FIGURE37













where












where
















andobtain








Wehave
























distributionweget




FIGURE38





fork=16




TABLE111


t2=ndash5353hours



Furthermorewefind

FIGURE39

thenwecompute



Forγ=05wehave




TABLE112



(Figure40)are

Wecomputenmin





FIGURE40
























or since λ0 = 1t0 = 000078 λ1 = 1t1 =








PROBLEMS








































formulasbecome

or




















PROBLEMS










form







































where



ANSWERSANDSOLUTIONS

I RANDOMEVENTS

















210

211

21221303

214

215


217

218



221




31

32



35














for


320321



Formlen2


Forn3lem2

Forn3gem2


weobtain


FIGURE41



weshallhave





49




419

420421

422

423

424

425P(A)=P(B)=P(C)=



428

429

430

431






510035205553pk= pkj542(rR)255112656







523


525p+q=1q= pp=

















wherebeingthe




61




Consider








615





71









marksmanfails











81(a)094=0656(b)094+4middot01middot093=0948

82 (a) (b)

83






TABLE114

815p=1ndash09510=04816p=1ndash095=041817



821(a) (b)3p4ndash4p3+ =0p=0614822

823 8240784825The200wones(R61=0394R102=0117)82606482702816828

829 830Werequire

831 We require

832P4 0=03024P4 1=04404P4 2=02144P4 3 = 00404P4 4 =00024

833026834015983595144



91p=P5221+2P5320=5024392p=P3111+P3210+P3120=0245

93(a) (b)

94 95

96 (a) (b)

97(a)





Thus







generatingfunction









Forn=10m=20









or


TABLE115



thatis

II RANDOMVARIABLES










TABLE120



111

112113

114

115(a)σ(b) (c)


p)1m(c)

117 (a) 10 (b) where

118 (a) (b) (c)

119


1111p=



1114

1115








129

1210

1211













1230


since


131M[X]=aD[X]= E=

132M[X]=0D[X]=

133

134 135

136 137M[X]=D[X]=m+1

138 139M[X]=0D[X]=2

1310 M[X]=(α+1)βD[X]=β2(α+1)1311

1312



1313

1314Usetherelation








1320

1321M[X]=0D[X]=

1322 wheremj=M[Xj]

1323 where

14 POISSONrsquoSLAW



144p=endash05asymp061145(1)095958(2)095963146091470143

148 14904

1410Sk=








155 156SeeTable122

TABLE122



1518





166

167E(u)= mk=k

168

169 wherev=u2hand


1610





1613







pointsbylweobtain

173p=015174


177

178

179

1710



181


182 (a) A = 20 (b)



TABLE123


(c)M[X]=1947M[Y]=0504

186





1812



18161817

1818


1820

1821



191192

193

194195

196

197


199



1912

1913

1914

1915

19161917

1918

wherep2=0196p3=0198p4=0148p5=0055q=0403

1919P= [Φ(k)]2

19201921

1922




201

202For|x|leR|y|leR



204205





2010



2011

2012

2013M[X|y]=08y+149M[Y|x]=045xndash8625

2014


2016

2017

2018

2019



2114aπ212π(a2)213M[G]=41gD[G]=032g2


2110211111a218π21123π2113

2114


2118


2119


2120

2121


21262127

2128(a)267sqm(b)220sqm(c)10sqm2129

21302131

2132M[Z]=0D[Z]=2Δ2σ2

21332134



223

224

225

226

227

228

229(b)ifagt0then

ifalt0then

(c)2210Foranoddn

forevenn

2211

22122213

2214

22162217

2218

2219


2220

2221







2226

2227




233234



238

239

231023112312




2317


241242

243

where

244



where


TABLE124


247

248249


2412


2415

2416

24172418SeeTable125

TABLE125




253

254





2516

2517





2521199mg

2522

2523







266(a)(a)a=b=25 m(b)a=689mb=388mα=15deg




FIGURE42





courseoftheship2617


2619


FIGURE43

FIGURE44


Fromthiswefind


where



271Since






278H[X]=loga(05 )


2710




27162717loga1050andloga302718











ForcodeNo2


288


(b)SeeTable127

TABLE127

(c)SeeTable128

TABLE128


(a)



TABLE129

TABLE130






where


2816

where

VI THELIMITTHEOREMS


291









298


(b)satisfiedsince





where




2913Consider











307nasymp65308p=0943309675






3015











Consequently










3110

3111


thatis

3113








=0P=030853211

3212

32133214



32173218D[Y(20)]=1360cm23219

3220





2τ2[1+2α2(1ndash2α2τ2)]

32253226




3231

where






Answer



338For ge545m

339







3313

where




where


341

342343Denoting

wehave

344345

346



34103411



3412Since

then

3413Since

then

3414Since


3415

34163417Since


where

3418Since then


where



34213422

3423



3424Since then


3425

where







353

where




355


356

where








3515



origin




3519

3520

3521

3522



and

then

3524









andforτle0




where

365

where

366

367

where

where


3611

where

3612

where

3613 3614

where


3616 where


where

3618

where






36223623

3624

3625

3626


371

where


372Since





378


then

3710Since




371300424αsecndash13714

where

3715


38 MARKOVCHAINS



Perron formula



where


ForigtkAki(λ)=0

Forkgti


ForigtkAki(λ)=0

Forkgtiknem







m) =0foriltkForigek






where



where






Theeigenvaluesare

where




Ifλ2=1then




weget



Then




itfollowsthat

Since






thatis


itfollowsthat

Theseries





thatis


weobtain







3827





qwehave



391

392 393

where


394 395





hasthesolutions







hasthesolutions












where

3915


hasthesolutions






where

thusitfollowsthat




401

402


whicha1=x2




408


409





where



where


where





where

40174018

where




4020

where






417

418

419

4110



4114 =4831m =5331m =1075m =1250m4115

where


4116 =23m =107m4117







421(9236m10764m)




TABLE135


4226

















4314 =875 =1685 =1186kH=5 =00398an




4316




4317 = 8746 = 2471 = 8002 = 9490 gt 500 kH = 7













4325


4326 =1153 =2143 =1020kH=10 =043


where



where




441 =0609+01242EM00=03896M11=000001156 =1464

=05704 =00000169442 =0679+0124E =1450 =05639 =000001672The


443


445

446




TABLE136

448 =03548+006574x+000130x2 =00147 =00106 =000156

449


TABLE137


44114412


4414

4415 =ndash3924+1306x|εmax|=141








456





TABLE138












463

464

465466




(b)230sec4611



4613Denotingby


Consequently





4616By9percent4617

4618Since

then






















see8T9T14T












distribution




Bartlett M1953














BIBLIOGRAPHY


































































Index

Absorbingstate232







































Quantile49














Title Page

Copyright Page

Foreward

Contents

I Random Events










II Random Variables



































38 Markov Chains












Bibliography

Index



BernardRGelbaum











II RANDOMVARIABLES





14 Poissonrsquoslaw



















VI THELIMITTHEOREMS












38 Markovchains










ANSWERSANDSOLUTIONS


BIBLIOGRAPHY

INDEX

I RANDOMEVENTS





















PROBLEMS




FIGURE1














PROBABILITIES

BasicFormulas














andhence


PROBLEMS







































FIGURE2

FIGURE3





FIGURE4

FIGURE5


andtherefore





PROBLEMS















































PROBLEMS















































Analogously Then




TABLE1



Since



FIGURE6




Fromthiswefind










andfinally


or


PROBLEMS
































530Provethat












where




where




Wehave










PROBLEMS

























givenbytheformula

where















PROBLEMS










































FIGURE7












PROBLEMS













FIGURE8


























































thatis















Wehave



PROBLEMS
































II RANDOMVARIABLES











TABLE2

FIGURE9







TABLE3






TABLE4












FIGURE10








FIGURE11



PROBLEMS























(3)

(4)


(1)f(x)ge0

(2)

(3)

(4)












weobtain





F(x05)=12

Solvingtheequation





Thisimpliesthat








TABLE5



PROBLEMS




























BasicFormulas





orby











Consequentlywehave





Consequently





















whosesumgives

or



TABLE6



ForthedeviceB




PROBLEMS
















(a)forthedeviceA

(b)forthedeviceB

FIGURE12






































whichforevennare

where

andforoddn



Thus

(b)Since

then

(c)

whereConsequently






PROBLEMS











(Maxwellrsquoslaw)
























density



expectation

14 POISSONrsquoSLAW















FIGURE13


PROBLEMS
















inwhich

and





form

or



inwhich


inwhich








Fromthisweget


andfinally











PROBLEMS
























BasicFormulas
















thisleadsto

thatis









FIGURE14


Similarlyforxgt0












andthemoments


PROBLEMS


































weobtain


Consequently

Since

then



and





or

PROBLEMS
































thevariances

andthecovariances









independentif








Similarlywefindthat






ThevarianceofXis


FIGURE15

Thecovarianceis








PROBLEMS







TABLE7






FIGURE16








1815Showthat


(X1X2X3)is








BasicFormulas









where















SOLUTION(a)Since

where

then

Thisimpliesthat








inwhich



PROBLEMS




the plane


















FIGURE17









FIGURE18
























SOLUTION(a)Since




Similarly

and




weobtain


since






weobtain

or

Similarlywefindthat


andsimilarly


Similarly





Consequently

FIGURE19



PROBLEMS






















distributionlaw
















then








where




Therefore




Therefore





Then


where

Finally






FIGURE20




PROBLEMS





















































FIGURE21








where

FIGURE22

Finallywehave





and





FIGURE23





for0ltzlt14

forndash14ltzlt0

FIGURE24A

FIGURE24B







Therefore




PROBLEMS



























distributionlaw




























wehave




equals






and

Therefore


or








Then

whichimpliesthat


PROBLEMS

































where




FIGURE25





FIGURE26

FIGURE27






FIGURE28

















Consequently



PROBLEMS
















where





























where


TABLE8



where





wherekvθ=rσvσθ





where



where



PROBLEMS

































FIGURE29







FIGURE30
















givenbytheformulas



FIGURE31




where


TABLE9






where


where






andconsequently







thatis


TABLE10



TABLE11

Check







Thus


PROBLEMS




TABLE12


TABLE13








aregiveninTable14

TABLE14









FIGURE32








TABLE15



FIGURE33





















and








TABLE17


andforthesecondone




underconstraints



undertheconstraints

and



andtherfore

where



distributionlaw




Itfollowsthat


PROBLEMS







































TABLE18







Thisimpliesthat

















TABLE19







PROBLEMS


























precedingletters

TABLE20











VI THELIMITTHEOREMS


BasicFormulas














inequalities














aπ then

Since







Wehave



PROBLEMS









TABLE21



TABLE22











BasicFormulas




















obtain

or














PROBLEMS

































distributionlaws

















then






PROBLEMS

311Provethat



















isgivenintheform




















andforthevariance

Since









obtains


Therequiredvariance








PROBLEMS













X(t)


























3225Given














known







Fornormalfunctions















Since

wehave

















Then





weobtain

thatis



PROBLEMS







































differentiableis




















weseethat

Since




SOLUTIONSince

PROBLEMS















FindSΔ(ω)if





















FIGURE34












wehave






Inthiscase




















Wehave

where

Since

then

and



if




PROBLEMS
















358Doestheequation














wherekgehgt0


3514Given



















3523Giventhesystem






























U(t)then

where




where








where















SOLUTIONInthiscase











PROBLEMS

















































if



















where


Similarly









randomfunction



Consequently




where


PROBLEMS







if



















38 MARKOVCHAINS





















































andconsequently












Then









Weset


Fromthiswefindthat





















whichcanbewrittenas


Thisimpliesthat












PROBLEMS












where






































relation








where

















applicable



Sincebydefinition

wehave










fornge1and













whichleadsto


















PROBLEMS

































where





































thatis









Consequently



equation


where















Henceitfollowsthat










where















Since


















wefindthatthatis









where







Thus



weobtainthat

PROBLEMS










where



wherethepolynomials





equations

where













andconsideringthat


where

























BasicFormulas






where

TABLE23






TABLE24







TABLE25







TABLE26





TABLE27





Therefore

where(jnei)


Here

Finallyweget


TABLE28








TABLE29


TABLE30




PROBLEMS






TABLE31







TABLE32





where












TABLE33

TABLE34


where



TABLE35



TABLE36




BasicFormulas






forunknownσ

where




where





Fromthis















and


Ifm=0thenp1=0and

Ifm=nthenp2=1and



(the













Therandomvariable


Then

Solvingtheequation







hours

















TABLE37


TABLE38






FIGURE35








Fromthis















Inthiscase

and


088(10141768)for 12040(00530808)for 13


(077094)for 12(005067)for 13

PROBLEMS



TABLE40




TABLE41






TABLE42

























TABLE44





BasicFormulas




















respectively
















TABLE46




TABLE47





TABLE48





Wefindthat






TABLE49






TABLE50


Computing








TABLE51



where

wefind


TABLE52










TABLE53



TABLE54


Determining




TABLE55




andY


TABLE56



TABLE57






PROBLEMS



TABLE58


TABLE59




TABLE60


TABLE61



TABLE62








TABLE63



TABLE64






TABLE65





TABLE66



TABLE67


TABLE68



TABLE69






TABLE70





TABLE71





TABLE72




TABLE73







where



where



TABLE75


FIGURE36

TABLE76





TABLE77



TABLE78






TABLE79



where



TABLE80



TABLE81







TABLE82



BasicFormulas





where































TABLE83



where












TABLE84

If





TABLE85

Weobtain






TABLE86



where




become





TABLE87




TABLE88








TABLE89

Weobtain


wefind



TABLE90





TABLE91









become


or


TABLE92




TABLE93

Weobtain




andget


TABLE94


andobtain


givenbytheformulas

Wehave


orby




TABLE95


TABLE96




TABLE97

weobtion









PROBLEMS


TABLE98







TABLE99



TABLE100





TABLE101




TABLE102






TABLE103



TABLE104




TABLE105




stream

TABLE106



TABLE107




TABLE108

If



where


TABLE109




TABLE110

Iftherelation



BasicFormulas











where







then



then







where



where







where



Ifnle01Np1lt01then



FIGURE37













where












where
















andobtain








Wehave
























distributionweget




FIGURE38





fork=16




TABLE111


t2=ndash5353hours



Furthermorewefind

FIGURE39

thenwecompute



Forγ=05wehave




TABLE112



(Figure40)are

Wecomputenmin





FIGURE40
























or since λ0 = 1t0 = 000078 λ1 = 1t1 =








PROBLEMS








































formulasbecome

or




















PROBLEMS










form







































where



ANSWERSANDSOLUTIONS

I RANDOMEVENTS

















210

211

21221303

214

215


217

218



221




31

32



35














for


320321



Formlen2


Forn3lem2

Forn3gem2


weobtain


FIGURE41



weshallhave





49




419

420421

422

423

424

425P(A)=P(B)=P(C)=



428

429

430

431






510035205553pk= pkj542(rR)255112656







523


525p+q=1q= pp=

















wherebeingthe




61




Consider








615





71









marksmanfails











81(a)094=0656(b)094+4middot01middot093=0948

82 (a) (b)

83






TABLE114

815p=1ndash09510=04816p=1ndash095=041817



821(a) (b)3p4ndash4p3+ =0p=0614822

823 8240784825The200wones(R61=0394R102=0117)82606482702816828

829 830Werequire

831 We require

832P4 0=03024P4 1=04404P4 2=02144P4 3 = 00404P4 4 =00024

833026834015983595144



91p=P5221+2P5320=5024392p=P3111+P3210+P3120=0245

93(a) (b)

94 95

96 (a) (b)

97(a)





Thus







generatingfunction









Forn=10m=20









or


TABLE115



thatis

II RANDOMVARIABLES










TABLE120



111

112113

114

115(a)σ(b) (c)


p)1m(c)

117 (a) 10 (b) where

118 (a) (b) (c)

119


1111p=



1114

1115








129

1210

1211













1230


since


131M[X]=aD[X]= E=

132M[X]=0D[X]=

133

134 135

136 137M[X]=D[X]=m+1

138 139M[X]=0D[X]=2

1310 M[X]=(α+1)βD[X]=β2(α+1)1311

1312



1313

1314Usetherelation








1320

1321M[X]=0D[X]=

1322 wheremj=M[Xj]

1323 where

14 POISSONrsquoSLAW



144p=endash05asymp061145(1)095958(2)095963146091470143

148 14904

1410Sk=








155 156SeeTable122

TABLE122



1518





166

167E(u)= mk=k

168

169 wherev=u2hand


1610





1613







pointsbylweobtain

173p=015174


177

178

179

1710



181


182 (a) A = 20 (b)



TABLE123


(c)M[X]=1947M[Y]=0504

186





1812



18161817

1818


1820

1821



191192

193

194195

196

197


199



1912

1913

1914

1915

19161917

1918

wherep2=0196p3=0198p4=0148p5=0055q=0403

1919P= [Φ(k)]2

19201921

1922




201

202For|x|leR|y|leR



204205





2010



2011

2012

2013M[X|y]=08y+149M[Y|x]=045xndash8625

2014


2016

2017

2018

2019



2114aπ212π(a2)213M[G]=41gD[G]=032g2


2110211111a218π21123π2113

2114


2118


2119


2120

2121


21262127

2128(a)267sqm(b)220sqm(c)10sqm2129

21302131

2132M[Z]=0D[Z]=2Δ2σ2

21332134



223

224

225

226

227

228

229(b)ifagt0then

ifalt0then

(c)2210Foranoddn

forevenn

2211

22122213

2214

22162217

2218

2219


2220

2221







2226

2227




233234



238

239

231023112312




2317


241242

243

where

244



where


TABLE124


247

248249


2412


2415

2416

24172418SeeTable125

TABLE125




253

254





2516

2517





2521199mg

2522

2523







266(a)(a)a=b=25 m(b)a=689mb=388mα=15deg




FIGURE42





courseoftheship2617


2619


FIGURE43

FIGURE44


Fromthiswefind


where



271Since






278H[X]=loga(05 )


2710




27162717loga1050andloga302718











ForcodeNo2


288


(b)SeeTable127

TABLE127

(c)SeeTable128

TABLE128


(a)



TABLE129

TABLE130






where


2816

where

VI THELIMITTHEOREMS


291









298


(b)satisfiedsince





where




2913Consider











307nasymp65308p=0943309675






3015











Consequently










3110

3111


thatis

3113








=0P=030853211

3212

32133214



32173218D[Y(20)]=1360cm23219

3220





2τ2[1+2α2(1ndash2α2τ2)]

32253226




3231

where






Answer



338For ge545m

339







3313

where




where


341

342343Denoting

wehave

344345

346



34103411



3412Since

then

3413Since

then

3414Since


3415

34163417Since


where

3418Since then


where



34213422

3423



3424Since then


3425

where







353

where




355


356

where








3515



origin




3519

3520

3521

3522



and

then

3524









andforτle0




where

365

where

366

367

where

where


3611

where

3612

where

3613 3614

where


3616 where


where

3618

where






36223623

3624

3625

3626


371

where


372Since





378


then

3710Since




371300424αsecndash13714

where

3715


38 MARKOVCHAINS



Perron formula



where


ForigtkAki(λ)=0

Forkgti


ForigtkAki(λ)=0

Forkgtiknem







m) =0foriltkForigek






where



where






Theeigenvaluesare

where




Ifλ2=1then




weget



Then




itfollowsthat

Since






thatis


itfollowsthat

Theseries





thatis


weobtain







3827





qwehave



391

392 393

where


394 395





hasthesolutions







hasthesolutions












where

3915


hasthesolutions






where

thusitfollowsthat




401

402


whicha1=x2




408


409





where



where


where





where

40174018

where




4020

where






417

418

419

4110



4114 =4831m =5331m =1075m =1250m4115

where


4116 =23m =107m4117







421(9236m10764m)




TABLE135


4226

















4314 =875 =1685 =1186kH=5 =00398an




4316




4317 = 8746 = 2471 = 8002 = 9490 gt 500 kH = 7













4325


4326 =1153 =2143 =1020kH=10 =043


where



where




441 =0609+01242EM00=03896M11=000001156 =1464

=05704 =00000169442 =0679+0124E =1450 =05639 =000001672The


443


445

446




TABLE136

448 =03548+006574x+000130x2 =00147 =00106 =000156

449


TABLE137


44114412


4414

4415 =ndash3924+1306x|εmax|=141








456





TABLE138












463

464

465466




(b)230sec4611



4613Denotingby


Consequently





4616By9percent4617

4618Since

then






















see8T9T14T












distribution




Bartlett M1953














BIBLIOGRAPHY


































































Index

Absorbingstate232







































Quantile49














Title Page

Copyright Page

Foreward

Contents

I Random Events










II Random Variables



































38 Markov Chains












Bibliography

Index

problems in probability theory, mathematical statistics and theory of random functions

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