1. (a) Suppose that X and Y are two independent continuous random variables with finite variances. Show that Var(Y|X = x) does not depend on r e R. [4] (b) Now let Y = AX + BX3, where the three random variables X ~ N(1,1) and A, B ~ U(-1,1) are all independent. You may assume without proof that X and Y are jointly continuous and that BI Xk holds for all k e N. i. Show that E[Y|X] = 0 and compute E[Y]. [3] ii. Compute corr(X,Y). (TYPE:) From this result, what can you conclude on whether X and Y are independent? Justify your conclusion [4] iii. Compute Var(Y|X = x). (TYPE:) From this result, what can you conclude on whether X and Y are independent? Justify your conclusion. [4]