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)1Thepartitiontheoremsays thatifBnis a partitionof thesamplespacethenE[X] =XnE[XjBn]P(Bn)Now supposethatXandYarediscreteRV' in therangeofYthenY=yisa event withnonzeroprobability, so we canuseit as theBin theabove.
The partition theorem says that if Bn is a partition of the sample space then E[X] = X n E[XjBn]P(Bn) Now suppose that X and Y are discrete RV’s. If y is in the range of Y then Y = y is a event with nonzero probability, so we can use it as the B in the above.
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