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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,trialsareconductedwheretworandomvariablesareobservedsimultaneouslyinordertodeterminenotonlytheirindividualbehaviorbutalsothedegr
2、eeofrelationshipbetweenthem.(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)Thedistribut
3、ionofprobabilityisspecifiedbylistingtheprobabilitiesassociatedwithallpossiblepairsofvaluesxandy,eitherbyformulaorinatable.Werefertothefunctionp(x,y)=P(X=x,Y=y)andthecorrespondingpossiblevalues(X,Y)asthejointprobabilitydistribution(联合分布)ofXandY.XYXy1y2…yj…x1p11
4、p12…p1jx2p21p22…p2j…Xipi1pi2…pij……Theysatisfy,wherethesumisoverallpossiblevaluesofthevariable.Example5.1.1CalculatingprobabilitiesfromadiscretejointprobabilitydistributionLetXandYhavethejointprobabilitydistribution.XY010120.10.20.40.20.10(a)Find;(b)Findtheprob
5、abilitydistributionoftheindividualrandomvariableX.Solution(a)Theeventiscomposedofthepairsofvalues(l,1),(2,0),and(2,l).Addingtheircorrespondingprobabilities(b)SincetheeventX=0iscomposedofthetwopairsofvalues(0,0)and(0,1),weaddtheircorrespondingprobabilitiestoobt
6、ain.Continuing,weobtainand.Insummary,,andistheprobabilitydistributionofX.Notethattheprobabilitydistributionofappearsinthelowermarginofthisenlargedtable.TheprobabilitydistributionofYappearsintheright-handmarginofthetable.Consequently,theindividualdistributionsa
7、recalledmarginalprobabilitydistributions.（边缘分布）XY01pX(x)0120.10.20.40.20.100.30.60.1pY(y)0.60.41.0Fromtheexample,weseethatforeachfixedvalueofx,themarginalprobabilitydistributionisobtainedas,wherethesumisoverallpossiblevaluesofthesecondvariable.Continuing,weobt
8、ain.Example3.5.3Supposethenumberofpatentapplications（专利申请）submittedbyacompanyduringa1-yearperiodisarandomvariablehavingthePoissondistributionwithmean,()andthevariousapplicationsind
