STAT 408: Homework 6 1. Suppose the covariances between X] and X2 is 5, between X] and X's is 4, and between X2 and X3 is 0. Moreover, the variances of X1, X2, Xs are 49, 25,9, respectively. (a) Write the 3x3 covariance (E) and correlation (R) matrices of the random vector X = (X1, X2, X3). For the next parts use the fact that for any scalars a1, a2, as: Var(a, X1 + a2 X2 + asX3) = a; Var(X1) + a; Var(X2) + a; Var(X3) + 2aja2Cov(X1, X2) + 2aja;Cov(X1, X3) + 2aza;Cov(X2, X3). (b) Use the above formula and compute the (numerical) value of Var(al X1 + 02X2 + asX3) for the following choices of a = (a1, 02, 03): 1. a = (2, -2, 0). 2. a = (2, -1,5), 3. a = (-1, 1, 2). 4. a = (-81, -82, 1). In the last case, , and 8, are unknown. Explain why you or someone might be interested in choosing them so that the variance of the relevant linear combination of the three random variables is minimized. (Hint: Think in terms of prediction, least squares method,....). (c) Derive the normal equations for minimizing Q(b1, b2) = Var(X3 - b1X1 - bzX2), assuming that the random variables have means equal to zero. (d) Write the equations in (c) in matrix form first, then solve it for by and by. (e) (Bonus) Compute the inverse of the covariance matrix E in (a)