1.2 (Generalization of Problem 1.1.) Suppose that X1, X2, ... is a sequence of ran- dom variables with E(X?)

1.2 (Generalization of Problem 1.1.) Suppose that Xl , X2, ... is a sequence of ran- dom variables with E(Xt2) < oo and E(Xt) = IL. a. Show that the random variablef(X1, ... , Xn) that minimizes the conditional mean squared error, f (Xl, . .. , n f(Xl, ... , xn) = E[Xn+lIXl, ... , xn]. b. Deduce that the random variable f (Xl, ... , Xn) that minimizes the uncondi- tional mean squared error, f (Xl, ... , n is also f(Xl, ... , xn) = E[Xn+lIXl, ... , xn]. c. If Xl, X2, ... is iid with E(Xi2) < oo and EXI [L, where is known, what is the minimum mean squared error predictor of Xn+l in terms of Xl, ... , n d. Under the conditions of part (c) show that the best linear unbiased estimator , Xn is X + +Xn). (DL said to be an unbiased of in terms of Xl, . estimator of if EDL = for all V.) e. Under the conditions of part (c) show that X is the best linear predictor of Xn +1 that is unbiased for IL. is iid with E(x2) < 00 and [L, and if so f. If XX Xl -F +Xn, n l, 2, .. ., what is the minimum mean squared error predictor of Sn+l in terms of Sl, .