chapter 1 probability distribution

8/19/2019 Chapter 1 Probability Distribution

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Probability Distribution

Contents

Random variablesDiscrete Probability DistributionSpecial Discrete Probability Distribution: Binomial Distribution Poisson DistributionContinuous Probability DistributionSpecial Continuous Probability Distribution:

Normal Distribution

Chapter 1 : PROBABII!" DIS!RIB#!ION !opic 1$1 :!able o% Contents eave blan&


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Random Variables

A random variable is a quantitative variable whose values is determined by the outcome of arandom experiment. A random e'periment is a process that result in di%%erent outcomes (hen the e'periment is carriedout in the same manner several times$Example : Random Experiment: !estin) three electrical items$ Random variable: Number o% de%ective items$

et N denotes non de%ective item and D denotes de%ective item$

In this example the random variable is denoted b X.X can assumes any of the four possible values !"!#!$ . Random variables are represented by capital alphabets such as X and % while their valuesare represented by lower case alphabets such as x and y.

Chapter 1 : PROBABII!" DIS!RIB#!ION !opic 1$* : Random variables eave blan&

Outcomes of testing 3 electrical items Number of defective items, X

NNN 0NND 1

NDN 1

NDD 2

DNN 1

DND 2

DDN 2

DDD 3


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Chapter 1 : PROBABII!" DIS!RIB#!ION !opic 1$* : Random variables eave blan&

Random variables are classified accordin& to the set of values that they can ta'e.

(iscrete random variable assumes values that can be counted.

)ontinuous random variable assumes values contain in one or more intervals.+'amples o% discrete random variables: Number o% times a machine brea&s do(n in a month, Numbero% accidents occur in a %actory yearly, Number o% appointments scheduled in a month to see aconsultant$+'amples o% continuous random variables: !ime ta&en by a )ara)e to service a car, the resistance o%an electrical component, len)th o% iron bars produced by a machine$


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*he +ean and ,tandard (eviation of a (iscrete Random Variable

*he mean of a discrete random variable! denoted by -! is the value that is expected tooccur if an experiment is repeated a lar&e number of times. *he mean is also called the expected value and is denoted by EX/.

*he standard deviation of a discrete random variable! denoted by 0! measures thespread of its probability distribution.

1et X be a discrete random variable with probability distribution px/ .

Chapter 1 : PROBABII!" DIS!RIB#!ION !opic 1$- : Probability Distributions o% a discrete random variableseave blan&

*

**** 'p''p'

''p

σ=σ

µ−∑=µ−∑=σ

∑=µ

:deviationStandard

)( )( )( :Variance

)( :Mean


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Example*he probability distribution of X! the number of defective li&ht bulbs purchased by ashop'eeper is shown below.

2ind the mean and the standard deviation of the number of defective li&ht bulbs purchasedby the shop'eeper.

Chapter 1 : PROBABII!" DIS!RIB#!ION !opic 1$- : Probability distribution o% a discrete random variableseave blan&

Number o% de%ective bulbs, . / 1 * - 0

P.2'3 /$41 /$/4 /$15 /$/6 /$/1

' / 1 * - 0

P'3 /$41 /$/4 /$15 /$/6 /$/1

' P'3 / /$/4 /$-* /$16 /$/0 7 ' P'32/$68

'* P'3 / /$/4 /$50 /$06 /$15 7 '* P'321$-*

9918$/98-5$/:deviation Standard ==σ

9836.0)(,var 222

=−∑= µ σ x p xiance68$/3''p,mean =∑=µ9836.0)58.0(32.1,var 22 =−=σ iance


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,pecial (iscrete 3robability (istribution

*he 4inomial (istribution

!he binomial distribution is applied to e'periments that satis%y the conditions o% abinomial e'periment$

Conditions o% a Binomial +'periment

!he e'periment consists o% a %i'ed number o% trials, n$ +ach trial results in one o% t(o outcomes classi%ied as a success and a %ailure $

!he trials are independent$!he probability o% success, p must be the same in each trial$ !heprobability o% %ailure in a sin)le trial is denoted by ; and ; 2 1 < p$

Some terminolo)y:

A trial is an action (hich results in one o% several outcomes

=hen one trial does not a%%ect the outcome o% another trial, they are said to be independent$

Chapter 1 : PROBABII!" DIS!RIB#!ION !opic 1$0 : Binomial Distribution eave blan&


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Examples of the 4inomial Experiment

+'ample 1$

!hro( a dice 0 times and record the random variable . , the number o% times a 5 isobtained$!he conditions o% a binomial e'periment are satis%ied since!here is a %i'ed number o% trials, n20 thro( a die 0 times3+ach trial has t(o outcomes$ )ettin) a 5 is a success and any other outcome is a %ailure3

!he trials are independent$ !he result o% a thro( does not a%%ect the outcome o% the ne't thro(3$!he probability o% )ettin) a 5 is the same %or all the thro(s, p21>5

+'ample *$Suppose a multiple choice ;ui? has */ ;uestions and each ;uestion has 0 possibleans(ers$ et .2 number o% correct ans(ers obtained by a student (ho ans(er all the */;uestions$!here is a %i'ed number o% trials, n2*/+ach trial has t(o outcomes$ )ettin) a correct ans(er is a success and a (ron) ans(er is a %ailure3!he trials are independent$ !he result o% a ;uestion does not a%%ect the result o% the other ;uestions3$!he probability o% )ettin) a correct ans(er in each ;uestion is the same, p2 @ $

Chapter 1 : PROBABII!" DIS!RIB#!ION eave blan&!opic 1$0 : Binomial Distribution


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*he 4inomial (istribution

!he discrete random variable, . that represents the number o% successes in n trials o% a binomiale'periment is called a binomial random variable$

I% . %ollo(s a binomial distribution, (e (rite X54in n!p/ (here p is the probability of a success in asin&le trial and n is the fixed number of trials.

!he probability o% ' successes in n trials is )iven by

P.2'3 can also be obtained %rom the table o% cumulative binomial probabilities$

Chapter 1 : PROBABII!" DIS!RIB#!ION eave blan&!opic 1$0 : Binomial Distribution

.1!0 and 1)2)...(2)(1(! with

)!(!

! Recall

.,...2,1,0for

)1()(

≡−−=

−=

=

−

=== −−

nnnn

xn x

nC

n x

p p xnq pC x X P

x

n

xn x xn x x

n


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*he +ean and ,tandard (eviation of the 4inomial (istribution

or a binomial e'periment (ith n trials and probability o% success, p in a sin)letrial, the mean and standard deviation o% the binomial distribution is )iven by

Chapter 1 : PROBABII!" DIS!RIB#!ION eave blan&

np;:deviation Standardnp;

*

:Bariance

np:Cean

=σ=σ

=µ

np;:deviation Standard

np;* :Bariance

np:Cean

=σ

=σ

=µ

!opic 1$0 : Binomial Distribution


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Example

!he probability that a patient recovers %rom a rare %lu disease is /$0$ It is &no(n that */ peoplehave contracted this disease, (hat is

a3the probability that e'actly 6 survive

b3the probability that at least - survive

c3the probability that %rom - to 8 survive

d3the probability that at most - do not survive

e3 the e'pected number o% survivors and thestandard deviation

Note:!he problem can be modeled by a binomial distribution becausei3!here are * outcomes: success2a patient survive recover3 %ailure2a patient did not survive$

ii3i'ed number o% trials, n2 */ patientsiii3!he trials are independent and probability o% success is the same %or each trial patient3, p2/$0

eave blan&!opic 1$0 : Binomial DistributionChapter 1 : PROBABII!"DIS!RIB#!ION


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2indin& 3robabilities 6sin& )umulative 4inomial 3robabilities *able

!able 1 )ive the cumulative probabilities

%or selected values of n and p$

+'ample 1#se !able 1 to %ind P.E*3 (here n26 p2/$/5

ind column mar&ed p2/$/5$

ind the ro( mar&ed n26 and r2*Fence obtain the value /$/-19$ So P.E*32/$/-19$

R61E, for usin& 4inomial and 3oisson *able 3 X 7 r /

a3P . E r 3 2 table

b3 P . G r 3 2 1 H P . E r 1 3 c3 P r G . G s 3 2 P . E r 3 H P . E s 1 3 d3 P . 2 r 3 2 P . E r 3 H P . E r 1 3

Chapter 1 : Study (or& order instructions !opic 1$* : Computer system unit components eave blan&

∑=

−−

=≥

n

r x

xn x p p

x

nr X P )1()(


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+'ample *

Consider .JBin 1/, /$/43 and usin) !able 1, (e can %ind the %ollo(in)probabilities:

a3P.E-3b3P.K-3c3P.L-3d3P.G-3e3P.2-3%3P1L.G 03



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*he 3oisson 3robability(istribution

Suitable to model data that represent the number o% occurrenceso% a speci%ied event in a )iven unit o% time or space$

+'amples o% Poisson variable

a3number o% accidents per year in a %actory$

b3 number o% cars per hour passin) throu)h a brid)e$

c3 number o% %aults in a meter lon) o% cable

d3 number o% typin) errors made in one pa)e$

eave blan&!opic 1$0 : Poisson DistributionChapter 1 : PROBABII!"DIS!RIB#!ION


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*he 3oisson 3robability(istribution

I% an event is randomly scattered in time or space3 andI% an event is randomly scattered in time or space3 andhas a mean number o% occurrences, M in a )iven intervalhas a mean number o% occurrences, M in a )iven intervalo% timeor space3 and i% .2no o% occurrences in the )iveno% timeor space3 and i% .2no o% occurrences in the )iveninterval then (e (rite .JPoM3 $interval then (e (rite .JPoM3 $

!he probability that the number o% occurrence e;uals ' is!he probability that the number o% occurrence e;uals ' is)iven by)iven by

!he Poisson distribution hasean2 MMStandard deviation2 MM

P.2'3 can also be obtained %rom the table o% cumulative

Poisson probabilities$

eave blan&!opic 1$0 : Poisson DistributionChapter 1 : PROBABII!"DIS!RIB#!ION

MM 21

MM 2-

MM 25

'

p'3arious Poisson Distributions

( ) /,1,*,$$$$ '

'

e'.P'

=µ

== µ−


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)umulative 3oisson 3robabilities *able

!he tabulated values in !able * are the probabilities

%or various values o% m, the mean number o%occurrence o% the event per interval$

2indin& 3robabilities usin& )umulative 3oisson

3robabilities *able

Consider .JPo/$83$!o obtain P.E*3 usin) !able *:oo& up column m2/$8 and ro( r2* (e obtain/$191*Fence P.E*32/$191*


( ) /,1,*,$$$$'%or '

mer .P

r '

'm ==≥ ∑

∞

=

−


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eave blan&Chapter 1 : PROBABII!"DIS!RIB#!ION

!opic 1$0 : Poisson Distribution


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3oisson distribution as an approximation to the 4inomial distribution

=hen n is lar)e nK 6/3 and p is small pL/$13the binomial distribution .JBinn,p3 can be appro'imated usin)a Poisson distribution (ith the mean M2np$

!he appro'imation )ets better as n )ets lar)er and p )ets smaller$

+'ample $

A lar)e lot o% items is &no(n to contain 0Q de%ective items$ I% a sample o% 1// is randomly dra(n %rom

the lot, use the Poisson appro'imation to %ind the probability it (ill containa3 no de%ectiveb3 more than 6 de%ectives$

Solutionet . 2 number o% de%ective items in a sample o% 1// items$ .JBin1//, /$/03#sin) Poisson appro'imation, M21// ' /$/0 2 0 .JPo03

a3

b3 P.K63 2P.E532/$*109 usin) Poisson table3

eave blan&Chapter 1 : PROBABII!"DIS!RIB#!ION

!opic 1$0 : Poisson Distribution

( ) //18-$/

0e/.P/0 === −


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)ontinuous 3robability (istribution

A smooth curve describes the probability distribution o% acontinuous random variable$

!he probability distribution o% a continuous random variable,. is described by a mathematical %ormula %'3 &no(n as theprobability density %unction or simply density %unction$

!he %unction %'3 is the probability density %unction o% acontinuous random variable . i%

1$ %or all real values o% '$

*$

!he probability that . assumes values in the interval a to b is

)iven by

Note:!here is no probability attached to any sin)le value o% '$+'ample: P. 2 a3 2 /$

!opic 1$6 : Continuous Probability Distribution eave blan&

0≥)( x f

∫ ∞

∞−

= 1dx x f )(

∫ =


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*he 8ormal (istribution

One important continuous random variable is thenormal random variable$!he normal or aussian distribution has a probabilitydensity %unction

!he parameters o% the normal distribution are itsmean µ and standard deviation σ $I% the random variable . has a normal distribution(ith parameters M and , (e (rite . J N M, * 3$

2eatures of the 8ormal curve

1$Bell shaped curve and is symmetric about the

mean$*$!otal area under the curve is 1$-$!he curve never touches the ' a'is$0$!he mean, median and mode are e;ual$

!opic 1$5 : Normal Distribution eave blan&Chapter 1 : PROBABII!"DIS!RIB#!ION

( ) ∞


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*he 8ormal(istribution

!he shapeand locationo% the normalcurvechan)es asthe meanand standarddeviation

chan)e$



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*he ,tandard 8ormal (istribution

!o %ind Pa L . L b3, (e need to evaluate (here

!his inte)ral is di%%icult to evaluate$

!o simpli%y the evaluation o% the inte)ral, (e trans%orm

variable ' to ? (here $

T is &no(n as the standard normal variable$

!he probability distribution o% T is the standard normaldistribution (ith mean, M2/ and standard deviation 21 $ =e

(rite T J N /,1 3$

!opic 1$5 : !he Normal Distribution eave blan&Chapter 1 : PROBABII!"DIS!RIB#!ION

∫ b

a

dx x f )(

( ) ∞


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*he ,tandard 8ormal (istribution

!he probability %unction o% T is

2eatures of the standard normal curve

1$Bell shaped curve and is symmetric about ?2/$*$!otal area under the curve is 1$-$alues o% ? to the le%t o% the centre is ne)ative and tothe ri)ht o% the centre is positive$0$ean2/, standard deviation 21$

*he ,tandard 8ormal *able

Areas under the standard normal curve aretabulated in a standard normal table$

Fence to %ind Pa L . L b3, (e use standardnormal table$


( ) ∞


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*he ,tandard 8ormal *able

!able - )ives the values o% PTK?3 %or ?K/(hich is the area under the ? curve to theri)ht o% a particular value o% T$

!o %ind P T K 1$*0 3 %rom the !able -, (e )oto the ro( in the table (ritten 1$* and loo& atthe value in the column headed /$/0, soP T K 1$*0 3 2 /$1/46



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2indin& 3robabilities 6sin& *he ,tandard 8ormal *able

+'ample

1$ et T be a standard normal random variable$ ind the %ollo(in) :

a3PTLH1$563b3PTL1$563c3P1$8LTL *$-3d3 PH1$8LTL *$-3

*$ et T be a standard normal random variable$ ind the constant & such thata3 PT L &32/$10*- b3 PT K &32/$9/8* c3 PH/$6 L T L &32 /$68**



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3roblem ,olvin& involvin& *he 8ormal (istribution

+'ample

1$Suppose the len)ths o% iron rods produced by a %actory are normally distributed (ith a meano% 1*/ cm and a standard deviation o% 1/ cm$

a3 =hat is the probability that an iron rod (ill have a len)th (ithin 6 cm o% the meanb3 Rods shorter than & cm are reUected$ +stimate the value o% & i% 9 Q are reUectedc3 In a sample o% 6// rods, estimate the number havin) a len)th over 1*5cm


chapter 1 probability distribution

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