conditional expectation(条件期望讲义).pdf

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1、AConditionalexpectationA.1Reviewofconditionaldensities,expectationsWestartwiththecontinuouscase.Thisissections6.6and6.8inthebook.LetX;Ybecontinuousrandomvariables.WedenedtheconditionaldensityofXgivenYtobefX;Y(x;y)fXjY(xjy)=fY(y)ThenZbP(aXbjY=y)=fX;Y(xjy)dxaConditioningonY=yiscond

2、itioningonaneventwithprobabilityzero.Thisisnotdened,sowemakesenseoftheleftsideabovebyalimitingprocedure:P(aXbjY=y)=limP(aXbjjYyj<)!0+WethendenetheconditionalexpectationofXgivenY=ytobeZ1E[XjY=y]=xfXjY(xjy)dx1Wehavethefollowingcontinuousanalogofthepartitiontheorem.Z1E[Y]=E[Y

3、jX=x]fX(x)dx1Nowwereviewthediscretecase.Thiswassection2.5inthebook.Insomesenseitissimplerthanthecontinuouscase.Everythingcomesdowntotheveryrstdenitioninvolvingconditioning.ForeventsAandBP(AB)P(AjB)=P(B)assumingthatP(B)>0.IfXisadiscreteRV,theconditionaldensityofXgiventheeventBisP

4、(X=x;B)f(xjB)=P(X=xjB)=P(B)andtheconditionalexpectationofXgivenBisXE[XjB]=xf(xjB)x1ThepartitiontheoremsaysthatifBnisapartitionofthesamplespacethenXE[X]=E[XjBn]P(Bn)nNowsupposethatXandYarediscreteRV's.IfyisintherangeofYthenY=yisaeventwithnonzeroprobability,sowecanuseitastheBintheabov

5、e.Sof(xjY=y)isdened.Wecanchangethenotationtomakeitlooklikethecontinuouscaseandwritef(xjY=y)asfXjY(xjy).OfcourseitisgivenbyP(X=x;Y=y)fX;Y(x;y)fXjY(xjy)==P(Y=y)fY(y)Thislooksidenticaltotheformulainthecontinuouscase,butitisreallyadierentformula.IntheabovefX;YandfYarepmf's;inthecontin

6、uouscasetheyarepdf's.WiththisnotationwehaveXE[XjY=y]=xfXjY(xjy)xandthepartitiontheoremisXE[X]=E[XjY=y]P(Y=y)yA.2ConditionalexpectationasaRandomVariableConditionalexpectationssuchasE[XjY=2]orE[XjY=5]arenumbers.IfweconsiderE[XjY=y],itisanumberthatdependsony.Soitisafunctionofy.Inthisse

7、ctionwewillstudyanewobjectE[XjY]thatisarandomvariable.Westartwithanexample.Example:Rolladieuntilwegeta6.LetYbethetotalnumberofrollsandXthenumberof1'sweget.WecomputeE[XjY=y].TheeventY=ymeansthattherewerey1rollsthatwerenota6andthentheythrollwasasix.Sogiventhisevent,Xhasabinomialdistr

8、ibutionwithn=y1tri