--- Exercise 22 (#1.51, #1.53). Let X be a random n-vector having the multivariate normal distribution Nn(u, In). (i) Apply Cochran's theorem to show that if A2 = A, then XAX has the noncentral chi-square distribution x?(5), where A is an n x n symmetric matrix, r = rank of A, and d = u' Au. (ii) Let A be an n x n symmetric matrix satisfying A Ai, i 1,2. Show that a necessary and sufficient condition that XTA X and XTA2X are independent is A1 A2 = 0. Note. If X1, ..., Xk are independent and X; has the normal distribution N(Mi, 02), i = 1, ..., k, then the distribution of (X +...+XX)/o2 is called the noncentral chi-square distribution x (), where 8 = (u + ... + M*)/o2. When 8 = 0, x is called the central chi-square distribution. Exercise 28 (#1.57). Let U, and U, be independent random vari- ables having the X, (S) and Xn, distributions, respectively, and let F = (Ui)/(U22). Show that (i) E(F) = n2(ni+8) when n2 > 2; (ii) Var(F) = 2n][(n 70)2+(ma-2)(na +28)) when n2 > 4. Note. The distribution of F is called the noncentral F-distribution and denoted by Fn1,ng(8). ni(n2-2)