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1、5.RandomvectorsandJointProbabilityDistributions随机向量与联合概率分布5.1ConceptofJointProbabilityDistributions(1)DiscreteVariablesCase离散型Often,trialsareconductedwheretworandomvariablesareobservedsimultaneouslyinordertodeterminenotonlytheirindividualbehaviorbutalsothedegreeofr
2、elationshipbetweenthem.(X,Y)FortwodiscreterandomvariablesXandY,wewritetheprobabilitythatXwilltakethevaluexandYwilltakethevalueyasP(X=x,Y=y).Consequently,P(X=x,Y=y)istheprobabilityoftheintersectionoftheeventsX=xandY=y.(X=x,Y=y)------(X=x)∩(Y=y)Thedistributionofproba
3、bilityisspecifiedbylistingtheprobabilitiesassociatedwithallpossiblepairsofvaluesxandy,eitherbyformulaorinatable.Werefertothefunctionp(x,y)=P(X=x,Y=y)andthecorrespondingpossiblevalues(X,Y)asthejointprobabilitydistribution(联合分布)ofXandY.28XYXy1y2…yj…x1p11p12…p1jx2p21p
4、22…p2j…Xipi1pi2…pij……Theysatisfy,wherethesumisoverallpossiblevaluesofthevariable.Example5.1.1CalculatingprobabilitiesfromadiscretejointprobabilitydistributionLetXandYhavethejointprobabilitydistribution.XY010120.10.20.40.20.10(a)Find;(b)Findtheprobabilitydistributio
5、noftheindividual28randomvariableX.Solution(a)Theeventiscomposedofthepairsofvalues(l,1),(2,0),and(2,l).Addingtheircorrespondingprobabilities(b)SincetheeventX=0iscomposedofthetwopairsofvalues(0,0)and(0,1),weaddtheircorrespondingprobabilitiestoobtain.Continuing,weobta
6、inand.Insummary,,andistheprobabilitydistributionofX.Notethattheprobabilitydistributionofappearsinthelowermarginofthisenlargedtable.TheprobabilitydistributionofYappearsintheright-handmarginofthetable.Consequently,theindividualdistributionsarecalledmarginalprobabilit
7、ydistributions.（边缘分布）XY01pX(x)0120.10.20.40.20.100.30.60.1pY(y)0.60.41.0Fromtheexample,weseethatforeachfixedvalueofx,themarginal28probabilitydistributionisobtainedas,wherethesumisoverallpossiblevaluesofthesecondvariable.Continuing,weobtain.Example3.5.3Supposethenum
8、berofpatentapplications（专利申请）submittedbyacompanyduringa1-yearperiodisarandomvariablehavingthePoissondistributionwithmean,()andthevariousapplicati
