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1、Chapter5ContinuousRandomVariablesandVectorsWehaveseenintheIntermezzothatthereisaneedtogeneralisethenotionsofanexperimentandarandomvariable.Inthischapterwesuggestasetupwhichallowsustodothis.Asinthe?rsttwochapters,we?rstde?neexperiments,andafterthatrandomvari
2、ables.Thetheoryinthischapterisbuiltupsimilarlyasthetheoryinthe?rsttwochapters.5.1ExperimentsInthediscretetheory,thesamplespaceΩwaseither?niteorcountablyin?nite.Inthepresentcontext,thisnolongersu?cesandwetakeΩ=Rd,thatis,ourexperimentswillhaveoutcomesinRd,for
3、somed∈N.Anexperimentinthediscretesettingconsistedofasamplespaceandaprobabilitymassfunction,assigningacertainprobabilitytoeachoutcomeinthesamplespace.TheprobabilityofaneventAinthediscretesettingcouldthensimplybede?nedasthesumofallprobabilitiesofelementsinA.I
4、nthecurrentcontinuoussetting,thisisimpossible,butwecandosomethingsimilar,replacingsumsbyappropriateintegrals.Herearesomeexamplesofhowthiscouldwork.Aftertheexamples,wegivetheformalde?nitions.Example5.1.1(Choosinganarbitrarypointfromtheunitinterval).Supposewe
5、wanttomodelthechoiceofacompletelyrandompointintheinterval(0,1).Thereareuncountablymanypointsin(0,1),andhencewecannotlistitselements.Insteadofconcentratingontheeventthatthechosenpointisequaltoacertaingivenelementof(0,1),weconsidertheeventthatthechosenpointfa
6、llsintoasubintervalI?(0,1).ItisreasonabletoassumethatthisprobabilityshouldbeequaltothelengthofI.Writing
7、I
8、forthelengthofanintervalI,wetherefore94Chapter5.ContinuousRandomVariablesandVectorsde?netheprobabilityofIasP(I)=
9、I
10、.Youshouldthinkofthisnumberastheprob
11、abilitythatacompletelyrandompointintheunitintervalendsupinI.Anotherwayofformulatingthisassignmentofprobabilitiesisthefollowing.De?nethefunction1if012、tegral.Whenwewritethingsthisway,thereisnoneedtorestrictourselvestointervalscontainedin(0,1).Indeed,sincef(x)=0forallx/∈(0,1),thepartofanintervaloutsidetheunitintervaldoesnotcontributetotheprobabilityof
