Stat 330 Mathematical Statistics - Assignment 2 You need to use the cover sheet provided in Learn. Due on Feb 27 (Monday) 12pm to the drop boxes located across the hall from MC 4065/4066. 1. [15 marks] Suppose X and Y are continuous random variables with joint p.d.f. f (x, y) = kexy , 0 < x < y < and 0 otherwise. Find (a) [2 marks] the value of k (b) [2 marks] P (X + Y 1) (c) [4 marks] the marginal p.d.f. of X and the conditional p.d.f of X given Y = y. (d) [3 marks] the joint c.d.f of X and Y (e) [4 marks] the joint moment generating function of (X,Y) 2. [15 marks] Suppose X and Y are discrete random variables with joint p.m.f. f (x, y) = kq 2 px+y , x = 0, 1, . . . ; y = 0, 1, . . . ; 0 < p < 1, q = 1 p (a) [2 marks] Determine the value of k (b) [4 marks] Find the marginal p.m.f. of X and the marginal p.m.f. of Y (c) [2 marks] Are X and Y independent ? Explain. (d) [3 marks] Find the joint moment generating function of (X, Y) (e) [4 marks] Use the moment generating function to find cov(X, Y ) = E(XY )E(X)E(Y ) 3. [10 marks] Let Y | P oisson() where fY | (y|) = y ,y y! e and Gamma(, ) = 0, 1, 2, . . . , > 0 and f () = 1 1 e/ , () > 0. (a) [5 marks] Find the mean and variance of Y without deriving the distribution of Y . (b) [5 marks] Show that the marginal distribution of Y is a negative binomial if is an integer. 1