MATH2421: Probability | Dr. YU, Chi Wai ASSIGNMENT 3 (Deadline: 4pm on April 24th) 1. A box contains 3 white and 2 black balls. The white balls are labelled by 1, 2, and 3, and the black balls by 4 and 5. A ball is randomly picked from the box. Let X be the number shown on the picked ball, and Y = 1 if the picked ball is black; Y = 0 otherwise. Find a. P(Y = 1); b. P(X = 4, Y = 1); c. E(XY); d. Var (Y|X = 4). 2. The joint pmf of X and Y is given by Px, Y ( x, y ) = - xty 27 - for x = 0, 1,2; y = 1, 2, 3, and Px,y (x, y) = 0 otherwise. a. Find E (Y|X = x) for all x = 0,1, 2. b. Find E (3 + 0.2Y|X = 2). 3. Given that X|Y = y ~ Binomial(n, y) for any y E (0, 1) and Y ~ U(0, 1), find the marginal pmf of X. Hint: Px (x) = S_[Pxx (xly)fx(y)]dy. 4. Given the following joint pdf of X and Y fx,x (x, y) = $24xy, forx> 0,y> 0 and x ty 2Y). c. Find E (XY). d. Find E (X|Y = y) for 0