stochastic lecture 5

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Stochastic lecture 5

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Copyright Syed Ali Khayam 2009EE 801 Analysis of Stochastic SystemsMultiple Random VariablesMuhammad Usman IlyasSchool of Electrical Engineering & Computer ScienceNational University of Sciences & Technology (NUST)PakistanCopyright Syed Ali Khayam 2009 Inthislecture,wewillcover: JointDistributionsofMultipleRandomVariables FunctionsofMultipleRandomVariables MomentsofMultipleRandomVariables JointlyNormalRandomVariables SumsofRandomVariablesWhatwillwecoverinthislecture?Copyright Syed Ali Khayam 2009VectorRandomVariablesPairsofRandomVariables3Copyright Syed Ali Khayam 2009DefinitionofaJointlyDistributedRandomVariable Avectorrandomvariableisamappingfromanoutcomes ofarandomexperimenttoavector: nXZ S S domainRange or image of X4Copyright Syed Ali Khayam 2009DefinitionofaJointlyDistributedRandomVariableRandom ExperimentSXX, Y(x, y)2,: X YZ S S Image courtesy of www.buzzle.com/ 1234566 5 4 3 2 1SY5Copyright Syed Ali Khayam 2009JointDistributions Thejointcumulativedistributionfunction(cdf)ofapairofrvsXandYisdefinedas: Jointdistributionsarealsocalledcompounddistributions{ }{ }, ( , ) PrPr , , ,X YF a b X a Y bX a Y b a b= = - < < - < < 6Copyright Syed Ali Khayam 2009ExamplesofJointDistributions:ThrowingTwoDiceExperimentxypX,Y(x,y)(1,1)(6,6)7Copyright Syed Ali Khayam 2009ExamplesofJointDistributions:AJointGaussianDistribution8Copyright Syed Ali Khayam 2009JointDistributionsYX bXbYdd9Copyright Syed Ali Khayam 2009JointDistributionsbXbYdd,Integrating shaded region gives ( , )X YF b dYX10Copyright Syed Ali Khayam 2009PropertiesofJointCDFs F1: F2: F3:IfXandYarecontinuousrvsthenFX,Y(a,b)isalsocontinuous( ),0 , 1, ,X YF a b a b - < < ( ) ( )1 2 1 2 , 1 1 , 2 2and , ,X Y X Ya a b b F a b F a b 11Copyright Syed Ali Khayam 2009PropertiesofJointCDFs F4:( )( )and , or ,, 1, 0a bX Ya bX YF a bF a b-ab,Integrating shaded region gives ( , )X YF a bYX12Copyright Syed Ali Khayam 2009PropertiesofJointCDFs( ) and , , 1a bX YF a b abbYX13Copyright Syed Ali Khayam 2009PropertiesofJointCDFs( ) and , , 1a bX YF a b abbYX14Copyright Syed Ali Khayam 2009PropertiesofJointCDFs( ) and , , 1a bX YF a b abYX15Copyright Syed Ali Khayam 2009PropertiesofJointCDFs F5:( ) ( ), , aX Y YF a b F bbThese distributions are called marginal distributionsa( ) ( ), , bX Y XF a b F aYX16Copyright Syed Ali Khayam 2009PropertiesofJointCDFs( ) { }{ }{ } { }{ } ( ),lim , PrPrPr PrPra X YYF a b X Y bX Y bY b X XY b F b = = - - = - - - = - =17Copyright Syed Ali Khayam 2009PropertiesofJointCDFs( ) { } { } ( ),lim , Pr Pra X Y YF a b X Y b Y b F b = = =baYX18Copyright Syed Ali Khayam 2009PropertiesofJointCDFs( ) { } { } ( ),lim , Pr Pra X Y YF a b X Y b Y b F b = = =baYX19Copyright Syed Ali Khayam 2009PropertiesofJointCDFsba( ) { } { } ( ),lim , Pr Pra X Y YF a b X Y b Y b F b = = =YX20Copyright Syed Ali Khayam 2009PropertiesofJointCDFs F6:{ } , , , ,Pr and c ( , ) ( , ) ( , ) ( , )X Y X Y X Y X Ya X b Y d F b d F a d F b c F a c < < = - - +21Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddYX22Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYdd,Integrating shaded region gives ( , )X YF b dYX23Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddca{ }We want Pr c a X b Y d < < YX24Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddcaIntegrating the entire region gives FX,Y(b,d)YX25Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddcaFirst subtract F(a,d) from FX,Y (b,d)YX26Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddcaThen subtract FX,Y (b,c) from FX,Y (b,d)-FX,Y (a,d)YX27Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddcaFinally add FX,Y (a,c) to FX,Y (b,d)-FX,Y (a,d)-FX,Y (b,c){ } , , , ,Pr and c ( , ) ( , ) ( , ) ( , )X Y X Y X Y X Ya X b Y d F b d F a d F b c F a c < < = - - +YX28Copyright Syed Ali Khayam 2009JointProbabilityDensityFunction Thejointprobabilitydensityfunctionisdefinedas: Alternatively:2, ,( , ) ( , )X Y X Yf x y F x yx y= , ,( , ) ( , )yxX Y X YF x y f a b dadb- -= 29Copyright Syed Ali Khayam 2009PropertiesofJointCDFsbXbYddca{ } ,Pr c ( , )b dX Ya ca X b Y d f x y dydx < < = YX30Copyright Syed Ali Khayam 2009JointProbabilityDensityFunction Thejointprobabilitydensityfunctionmustsatisfythefollowingproperty: Moreover:, ( , ) 1X Yf x y dxdy - -= ,,( ) ( , )( ) ( , )X X YY X Yf x f x y dyf y f x y dx--==31Copyright Syed Ali Khayam 2009Homework ReadingAssignment:Examples4.1to4.9inthebook Homework Problem4.3inthetextbook Problem4.9inthetextbook32Copyright Syed Ali Khayam 2009IndependentRandomVariables TworandomvariablesXandYareindependentif Or:, ( , ) ( ) ( )X Y X YF x y F x F y =,,( , ) ( ) ( ) for discrete rvs( , ) ( ) ( ) for continuous rvsX Y X YX Y X Yp x y p x p yf x y f x f y==33Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables Conditionalprobabilityisanimportanttoolwhenanalyzingrandomvariables Ingeneral,weareinterestedinfindingtheprobabilitythatarandomvariableYtakesonacertainvalue(orarangeofvalues)giventhatanotherrandomvariableXhasalreadytakensomevalue Conditionalprobabilitycanbeexpressedintermsofthejointandmarginalprobabilities34Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariablesRandomExperimentWhatistheprobabilitythatthesumofthetwodiceisoddgiventhatX=2?XY35Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables Wecanextendthedefinitionofconditionalprobabilitytorandomvariables( | ) Pr{( ) | ( )}Pr{ }Pr{ }YF y x Y y X xY y X xX x= = ===36Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables Wecanextendthedefinitionofconditionalprobabilitytorandomvariables Whatswrongwiththisdefinition?( | ) Pr{( ) | ( )}Pr{( ) ( )}Pr{ }YF y x Y y X xY y X xX x= = ===37Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables Whatswrongwiththisdefinition? ThisdefinitionwillonlyworkfordiscretervsbecausePr{X=a}=0forcontinuousrvs Therefore,weneedtogeneralizethisdefinitiontoencompasscontinuousrvsPr{( ) ( )}( | )Pr{ }YY y X xF y xX x ===38Copyright Syed Ali Khayam 2009PropertiesofJointCDFsb0Pr{( ) ( )}( | ) limPr{ }YhY y x X x hF y xx X x h < +=< +ahYX39Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables Weneedtogeneralizethedefinitionofconditionalprobabilitytoencompassbothdiscreteandcontinuousrvs( )( )( )( )( )( )0, ,,Pr{( ) ( )}( | ) limPr{ }, ,,Yhy yx hX Y X Yxx hXXxyX YXY y x X x hF y xx X x hf x y dx dy hf x y dyhf xf x dxf x y dyf x+- -+- < +=< + = = 40Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables WecandifferentiatetheconditionalCDFtoobtaintheconditionalpdf( )( ), ,( | )yX YYXf x y dyF y xf x- =41( )( ), ,( | ) ( | ) X YY YXf x ydf y x F y xdy f x= =Copyright Syed Ali Khayam 2009ConditionalProbabilityforRandomVariables Inotherwords,ajointpdfcanbewrittenasaproductofconditionalandmarginalpdfs: Foradiscreterv,thesameconditioncanbestatedas:42( ) ( ), , ( | )X Y Y Xf x y f y x f x =( ) { } ( ), , Pr , ( | )X Y Y Xp x y X x Y y p y x p x = = = =Copyright Syed Ali Khayam 2009ConditionalExpectation Expectationofaconditionaljointdistributionisdefinedas43{ }( | ) continuous rvs|( | ) discrete rvsYYyyf y x dyE Y xyp y x-= Copyright Syed Ali Khayam 2009UsefulPropertiesofConditionalExpectation P1: P2:Forafunctionh(Y): ThekthmomentofYisgivenby44{ }{ } { } | E E Y x E Y ={ } { }{ }( ) ( ) | E h Y E E h Y x ={ } { } { }|k kE Y E E Y x =Copyright Syed Ali Khayam 2009Homework ReadingAssignment Example4.15to4.26inthetextbook45Copyright Syed Ali Khayam 2009VectorRandomVariablesMultipleRandomVariables46Copyright Syed Ali Khayam 2009JointDistributionofMultiplervs Mostoftheideasthatwehavestudiedsofarcanbedirectlyextendedtomorethantwojointlydistributedrandomvariables Jointpmfofndiscreterandomvariablesis: Conditionalpmfsareobtainedas47( ) { }1 2, , , 1 2 1 1 2 2, , , Pr , , ,nX X X n n np x x x X x X x X x = = = = ( ) ( )( )1 21 2 1, , , 1 2 11 2 1, , , 1 2 1, , , ,| , , ,, , ,nnnX X X n nX n nX X X np x x x xp x x x xp x x x----= Copyright Syed Ali Khayam 2009JointDistributionofMultiplervs Mostoftheideasthatwehavestudiedsofarcanbedirectlyextendedtomorethantwojointlydistributedrandomvariables Jointpmfofndiscreterandomvariablesis: Conditionalpmfsareobtainedas48( ) { }1 2, , , 1 2 1 1 2 2, , , Pr , , ,nX X X n n np x x x X x X x X x = = = = ( ) ( )( )1 21 2 1,

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