Let \(X\) and \(Y\) be two random variables. Denote the mean of \(Y\) given \(X=x\) by \(\mu(x)\) and the variance of \(Y\) by \(\sigma^{2}(x)\).

a. Show that the best (minimum MSPE) prediction of \(Y\) given \(X=x\) is \(\mu(x)\) and the resulting MSPE is \(\sigma^{2}(x)\). (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 \(\mu(X)\) and the resulting MSPE is \(E[Y-\mu(X)]^{2}=\) \(E\left[\sigma^{2}(X)\right]\).