X Eyxgtyg
Proof As we know, X and Y are independent if and only if fX;Y(x;y) = fX(x)fY(y) or, equivalently, fXjY(xjy)= fX(x) But then E(XjY =y)=åx xfXjY(xjy)=åx xfX(x)=E(X) 2 2 EE(g(X)jY)=E(g(X)) Proof Set Z = g(X) Statement (i) of Theorem 1 applies to any two rv’s Hence, applying it to Z and Y we obtain EE(ZjY)= E(Z) which is the same.
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