Lecture 10 - University of Texas at Austin

Lecture10: ConditionalExpectation1of17 Course:Theory of Probability ITerm:Fall2013 Instructor:Gordan ZitkovicLecture10 ConditionalExpectationThe definition and existence of conditional expectationFor eventsA,BwithP[B]>0, we recall the familiar objectP[A|B] =P[A B]P[B].We say thatP[A|B]theconditional probability ofA, isimportant to note that the conditionP[B]>0 is crucial. WhenXandYare random variables defined on the same probability space, we oftenwant to give a meaning to the expressionP[X A|Y=y], even thoughit is usually the case thatP[Y=y] =0. When the random vector (X,Y)admits a joint densityfX,Y(x,y), andfY(y)>0, the conceptof conditional densityfX|Y=y(x) =fX,Y(x,y)/fY(y)is introduced andthe quantityP[X A|Y=y]is given meaning via AfX|Y=y(x,y) this procedure works well in the restrictive case of absolutelycontinuous random vectors, we will see how it is encompassed bya general concept of a conditional expectation.

Jan 24, 2015 · When the random vector (X,Y) admits a joint density fX,Y(x,y), and fY(y) > 0, the concept of conditional density f XjY=y(x) = f, Y(x,y)/f (y) is introduced and the quantity P[X 2AjY = y] is given meaning via R A f XjY=y(x,y)dx. While this procedure works well in the restrictive case of absolutely continuous random vectors, we will see how it is ...

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