Problem 1. Let X1 a12 . . . a 2,2 . . . X2 a21 a 2n x = A = an, 1 an, 2 . . ann where a;; are constants and X1, X2, ..., Xn ~ N(0, 1) are independent and identically distributed standard normal random variables. (a) What is the covariance matrix _ for x? (Recall the (i, j)-entry of E is CoV[x;, x;]. (b) Show that E[x Ax] = tr(A) := @1,1 + a2,2+ ... + ann (hint: write out the expression for x Ax as a sum over the entires of A) (c) Suppose A is diagonal; i.e. a;; = 0 for all i # j. Compute the variance of x Ax. You may use the fact that x? is a Chi-square random variable with one degree of freedom so that V[x? ] = 2