2 STAT 330: Mathematical Statistics Winter 2016 Assignment 2 Name: ID: Due on Feb 29th (Monday) 11:30am in drop boxes across the hall from MC 4065/4066. 1 1. [7 marks] Let f (x, y) = kxy 2 , if 0 x 2, 0 y 2 0, otherwise be the joint pdf of X and Y , where k is a positive constant. (a) Find k. (b) Find P (X Y ). (c) Find the marginal pdf of X and Y . (d) Find the joint cdf of X and Y . (e) Are X and Y independent? Explicitly state your reason. 2. [7 marks] Let f (x, y) = 3/2, if x2 y 1, 0 x 1 0, otherwise be the joint pdf of X and Y. (a) Find P (0 X 1/2). (b) Find P (X 1/2, Y 1/2). (c) Are X and Y independent? Explicitly state your reason. (d) Find the marginal pdf of X and Y . (e) Find the conditional pdf of X given Y = y. (f) Find E(X|Y = y). 3. [7 marks] Suppose (X1 , . . . , Xk ) M U LT (n, p1 , . . . , pk ). To answer the following questions, you may use the facts: for i = j, (1). the joint mgf of Xi and Xj is M(Xi ,Xj ) (t1 , t2 ) = (et1 pi + et2 pj + 1 pi pj )n . (2). (Xi , Xj ) M U LT (n, pi , pj ). (3). Xi BIN (n, pi ). (4). Xi + Xj BIN (n, pi + pj ). 2 (a) Using the joint mgf, show that Cov(Xi , Xj ) = npi pj , for i = j. pi (b) Show that Xi |Xj = xj BIN (n xj , 1pj ) for i = j. (c) For i = j, let T = Xi + Xj . Show that Xi |T = t BIN (t, pi pi j ). +p 4. [10 marks] Suppose X = (X1 , X2 )T BV N (, ), where = (1 , 2 )T and = 2 1 1 2 2 1 2 2 (a) Show that the joint mgf of X1 and X2 is 1 M (t1 , t2 ) = exp(T t + tT t) for t 2 2 . (Hint: First show that (x)T 1 (x)2tT x = (xt)T 1 (xt)2T ttT t, where x = (x1 , x2 )T ). 2 2 (b) Use mgf to show X1 N (1 , 1 ), and X2 N (2 , 2 ). (c) Find Cov(X1 , X2 ). (d) Use mgf to show that X1 and X2 are independent if and only if = 0. (e) Let A be a 2 2 nonsingular matrix and b be a 2 1 vector. Use the mgf to show that Y = AX + b BV N (A + b, AAT ). 5. [9 marks] Suppose X is a random variable following the standard normal distribution N (0, 1), Y is a random variable, and the conditional pdf of Y given X = x is (yx)2 1 fY |X (y|x) = e 2 , < y < , 2 where x is a real number. (a) Find the joint pdf of X and Y . (b) Dene Z = Y X. Find the joint pdf of X and Z. (c) Are X and Z are independent? Justify your answer. (d) Find Cov(X, Y ). (e) Find the marginal pdf of Y . 3