8. Let Y1 and Y2 be two unbiased statistics for a parameter 0. Assume Y1 and Y2 are independent and that Var(Y1) = 2Var(Y2). For two constants a and B let Z = aY1 + BY2. Find constants a and S such that Z is also an unbiased estimator for 0 and is in fact an MVUE. 9. * Suppose X and Y have a bivariate normal distribution with means /, and /2, respectively, and positive variances ox = 3 = 07. (a) Find the joint density of (X, Y) if we assume they are independent (b) Suppose X and Y are not independent and have correlation coefficient p. Find the joint density. (This is also known as the bivariate normal density.) (c) Suppose X and Y are as in (9b). Show that E(Y (x) is given by E(Y |x) = 0(2) = 12 + P OY OX (d) Calculate E($(X)) (e) Show that " p ( x ) = p203