Suppose that X and Y are two independent continuous random variables with finite variances. Show that Var(Y|X = x) does not depend on x 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 B Xk holds for all k N. i. Show that E[Y|X] = 0 and compute E[Y]. [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. [3] ii. Compute corr(X, Y). [TYPE:] From this result, what can you conclude on whether X and Y are independent? Justify your conclusion

(a) Suppose that X and Y are two independent continuous random variables with finite variances. Show that Var(Y|X = :c) does not depend on a: E R. [4] (b) Now let Y = AX + 3X3, where the three random variables X N N(1, 1) and A, B N U(1, 1) are all independent. You may assume without proof that X and Y are jointly continuous and that B J. Xk holds for all k; E N. i. Show that E[Y|X] = 0 and compute IE[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 = 3:). [TYPE:] From this result, what can you conclude on whether X and Y are independent? Justify your conclusion. [4]