Problem 1 (110%). Let Y = (Yi, Y2, Y3) ~ Na(0, E), where 0 = (0, 0, 0) and E = Set X = (X1, X2, X3) , where X1 = Yi + Yz, X2 = Y2, and X3 = Yi + Y2 + 3)3. Answer the following questions (you may assume that the inverse of the required matrices exist): Al (10 pts.) Find the joint distribution of X = (X1, X2, X3) . A2 (10 pts.) Find the moment generating function of (X1, X3). A3 (10 pts.) Find all the pairs that X, and X} are independent. B1 (10 pts.) Find the distribution of {(Y1 - Y2)' + (Y2 - Ya)' + (13 - Wi)?) + Zil Y?. Cl (10 pts.) Find Ci such that Y and Y - XC1 are independent. C2 (10 pts.) Find the joint distribution of Y - XC1. D1 (10 pts.) Find C2 such that | |Y - XC2|| = (Y -XC2) (Y - XC2) is minimized. D2 (10 pts.) Given a full rank 3 x 3 matrix Z, find Cs such that ||Y - ZCal| = (Y -ZC3) (Y - ZC3) is minimized. D3 (10 pts.) Find E(C;) and Var(C3). El (10 pts.) Given a 3 x 3 diagonal matrix W11 0 W = 0 W22 0 0 0 W33, where wit > 0 for i = 1, 2, 3. Find C4 such that | Y-ZCallw = (Y-ZC.) W(Y-ZC4) is minimized (Z is defined in D2). E2 (10 pts.) Find E(C4) and Var(C4)