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1、Chapter2RandomVariablesandRandomVectorsItoftenhappensthatwedonotreallycareabouttheoutcomeofanexperimentitself,butratherweareinterestedinsomeconsequenceofthisoutcome.Forin-stance,agamblerisnotprimarilyinterestedinthequestionwhetherornotheadscomesup,butinstea
2、dinthefinancialconsequencesofsuchanoutcome.Hencethegamblerisinterestedinafunctionoftheoutcome,ratherthanintheoutcomeitself.Suchafunctioniscalledarandomvariableandinthischapterwedefineandstudysuchrandomvariables.2.1RandomVariablesSupposethatweflipacoinntimes.Ea
3、chtimetailscomesupweloseoneeuro,andwhenheadscomesupwewinoneeuro.Thesamplespacecorrespondingtothisexperimentcouldbe{−1,1}n,thesetofsequencesof−1sand1soflengthn.Howcanweexpressourfortuneafternflips?Writingω=(ω1,...,ωn)asusual,whereωi=1iftheithflipisahead,mywinn
4、ingsafternflipsareequaltonS(ω)=ωi.i=1ThusourwinningsisamappingS:Ω→R.Suchmappingsarecalledrandomvariables.Definition2.1.1.ArandomvariableXisamappingfromasamplespaceΩintoR.36Chapter2.RandomVariablesandRandomVectors0123Figure2.1:Asketchofthedistributionfunction
5、ofX.Typically,wedenoterandomvariableswithlettersneartheendoftheal-phabet,likeX,YandZ.Weareofteninterestedintheprobabilitythatarandomvariabletakescertainvalues.HenceweareinterestedinP({ω:X(ω)=x}),forallappropriatex.Weshalloftenwrite{X=x}for{ω:X(ω)=x}andP(X=x
6、)forP({ω:X(ω)=x}).Definition2.1.2.TheprobabilitymassfunctionofarandomvariableXisthefunctionpX:R→[0,1]givenbypX(x)=P(X=x).Definition2.1.3.ThedistributionfunctionofarandomvariableXisthefunctionFX:R→[0,1]givenbyFX(x)=P(X≤x).Example2.1.4.Supposethattherandomvaria
7、bleXtakesthevalue1withprob-ability1,thatis,P(X=1)=1.ThedistributionfunctionisthengivenbyFX(x)=1,forx≥1,andFX(x)=0forx<1.Notethatthisfunctioniscontinuousfromtherightbutnotcontinuousfromtheleftatx=1.Example2.1.5.ConsiderarandomvariableXwithP(X=1)=1/2,P(X=2)=
8、1/4andP(X=3)=1/4.ThedistributionfunctionofXisgivenby⎧⎪⎪0ifx<1,⎨1/2if1≤x<2,FX(x)=⎪⎪3/4if2≤x<3,⎩1ifx≥3,seeFigure2.1.Example2.1.6.Supposethatwetossafaircoinntimes.Thenumberofheadsisarandomvariablewhichwe
