14.8 Let X and Y be two random variables. Denote the mean of Y given X = x by m(x) and the variance of Y by s2(x).

a. Show that the best (minimum MSPE) prediction of Y given X = x is m(x) and the resulting MSPE is s2(x). (Hint: Review Appendix 2.2.)

b. Suppose X is chosen at random. Use the result in

(a) to show that the best prediction of Y is m(X) and the resulting MSPE is E[Y - m(X)]2 =

E [s2(X)].