X 2. (a) Let 1 O Y ~ N O, . Compute E[XY] without using Cov(X, Y) = O o or corr( X, Y ) = o. Hint: Rely on what you know about the conditional distribution of Y given X. [3] 1 R (b) Now let 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]