X 2. (a) Let Y ~ NO, . Compute E[XY] without using Cov(X, Y) = p or corr( X, Y) = p. Hint: Rely on what you know about the conditional distribution of Y given X. [3] X R~ N CO. ( 1 R (b) Now let Y R 1 where R takes values -1/2 and 1/2 each with probability 1/2. i. Compute /x = E[X'] and my = E[Y]. [2] ii. Compute E[XY] and hence obtain Cov(X, Y). Hence compute the covari- Var(X ) Cov(X, Y) ance matrix > = Cov(X, Y) Var(Y) [3] iii. Compute Var(Y X). Start with Var(Y X ) = Var(Y X, R = 1/2) P(R = 1/2) + Var(Y [X, R = -1/2) P(R = -1/2). [3] iv. [TYPE:] Decide whether the joint distribution of (X, Y) (which is not con- ditional on R) is Gaussian. Justify your decision carefully. [4]