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A Conditional expectation

A conditionaldensities,expectationsWe ; de nedtheconditionaldensity ofXgivenYto befXjY(xjy) =fX;Y(x; y)fY(y)ThenP(a X bjY=y) =ZbafX;Y(xjy)dxConditioningonY=yis conditioningonanevent withprobability notde ned,so we make senseof theleftsideabove by a limitingprocedure:P(a X bjY=y) = lim !0+P(a X bjjY yj< )We thende netheconditionalexpectationofXgivenY=yto beE[XjY=y] =Z1 1x fXjY(xjy)dxWe have thefollowingcontinuousanalogof [Y] =Z1 1E[YjX=x]fX(x)dxNow we somesenseitis thevery rstde eventsAandBP(AjB) =P(A\B)P(B)assumingthatP(B)> a discreteRV,theconditionaldensity ofXgiventheeventBisf(xjB) =P(X=xjB) =P(X=x; B)P(B)andtheconditionalexpectationofXgiv enBisE[XjB] =Xxx f(xjB)1 Thepartitiontheoremsays thatifBnis a partitionof thesamplespacethenE[X] =XnE[XjBn]P(Bn)Now supposethatXandYarediscreteRV' in therangeofYthenY=yisa event withnonzeroprobability, so we canuseit as theBin theabove.

Suppose that the random variables are discrete. We need to compute the expected value of the random variable E[XjY]. It is a function of Y and it takes on the value E[XjY = y] when Y = y. So by the law of the unconscious whatever, E[E[XjY]] = X y E[XjY = y]P(Y = y) By the partition theorem this is equal to E[X]. So in the discrete case, (iv) is ...

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