Continuous Random Variables and Vectors

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1、Chapter5ContinuousRandomVariablesandVectorsWehaveseenintheIntermezzothatthereisaneedtogeneralisethenotionsofanexperimentandarandomvariable.Inthischapterwesuggestasetupwhichallowsustodothis.Asinthefirsttwochapters,wefirstdefineexperiments,andafterthatrandomvari

2、ables.Thetheoryinthischapterisbuiltupsimilarlyasthetheoryinthefirsttwochapters.5.1ExperimentsInthediscretetheory,thesamplespaceΩwaseitherfiniteorcountablyinfinite.Inthepresentcontext,thisnolongersufficesandwetakeΩ=Rd,thatis,ourexperimentswillhaveoutcomesinRd,for

3、somed∈N.Anexperimentinthediscretesettingconsistedofasamplespaceandaprobabilitymassfunction,assigningacertainprobabilitytoeachoutcomeinthesamplespace.TheprobabilityofaneventAinthediscretesettingcouldthensimplybedefinedasthesumofallprobabilitiesofelementsinA.I

4、nthecurrentcontinuoussetting,thisisimpossible,butwecandosomethingsimilar,replacingsumsbyappropriateintegrals.Herearesomeexamplesofhowthiscouldwork.Aftertheexamples,wegivetheformaldefinitions.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.ContinuousRandomVariablesandVectorsdefinetheprobabilityofIasP(I)=

9、I

10、.Youshouldthinkofthisnumberastheprob

11、abilitythatacompletelyrandompointintheunitintervalendsupinI.Anotherwayofformulatingthisassignmentofprobabilitiesisthefollowing.Definethefunction1if0

12、tegral.Whenwewritethingsthisway,thereisnoneedtorestrictourselvestointervalscontainedin(0,1).Indeed,sincef(x)=0forallx/∈(0,1),thepartofanintervaloutsidetheunitintervaldoesnotcontributetotheprobabilityof