2.27 X and Z are two jointly distributed random variables. Suppose you know the value of Z, but not the value of X. Let X

= E(X  Z) denote a guess of the value of X using the information on Z, and let W = X - X denote the error associated with this guess.

a. Show that E(W ) = 0. (Hint: Use the law of iterated expectations.)

b. Show that E(WZ ) = 0.

c. Let X n = g(Z) denote another guess of X using Z, and V = X - X n denote its error. Show that E(V2) Ú E(W 2). [Hint: Let h(Z ) = g(Z) - E(X  Z), so that V = 3X - E(X  Z )4 - h(Z ). Derive E(V2).]