Transcription of Lecture 10 : Conditional Expectation
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Lecture 10 : Conditional Expectation
Lecture10 : ConditionalExpectationSTAT205 Lecturer:JimPitmanScribe:CharlessC. nitionof ConditionalExpectationRecallthe\undergra duate"de nitionof conditionalprobability associatedwithBayes'RuleP(AjB) P(A; B)P(B)For a discreterandomvariableXwe haveP(A) =XxP(A; X=x) =XxP(AjX=x)P(X=x)andtheresultingformulaf orconditionalexpectationE(YjX=x)=Z Y(!)P(dwjX=x)=RX=xY(!)P(dw)P(X=x)=E(Y1(X =x))P(X=x)We wouldlike to extendthisto handlemoregeneralsituationswheredensitie sdon'texistor wewant to conditiononvery\complicated" randomvariableYwithEjYj<1onthespace( ;F;P)andsomesub- - eldG Fwewill de netheconditionalexpectationas thealmostsurelyuniquerandomvariableE(YjG )whichsatis (YjG) (Y Z) =E(E(YjG)Z)forallZwhichare bounded andG-measurableForG= (X) whenXis a discretevariable,thespace is simplypartitionedinto disjoint sets =tGn. Ourde nitionforthediscretecasegivesE(Yj (X))=E(YjX)=XnE(Y1X=xn)P(X=xn)1X=xn=XnE( Y1Gn)P(Gn)1 Gnwhich is : escondition2 of De (Hint:showthattheconditionis satis ed forrandomvariablesof theformZ=1 GwhereG2 Cis a collectionclosed underintersectionandG= (C)theninvokeDynkin's ) WellDe (XjG)is uniqueup to almostsure Sketch:Supposethatbothrandomvariables^Ya nd^^Ysatisfyourconditionsforbeingthecond itionalexpectationE(YjX).
Lecture 10: Conditional Expectation 10-2 Exercise 10.2 Show that the discrete formula satis es condition 2 of De nition 10.1. (Hint: show that the condition is satis ed for random variables of the form Z = 1G where G 2 C is a collection closed under …
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