1. (10 points) Suppose Y has density function (a) (2 points) Find E[eY/2] 12, 0 < y < , f(y) O.W. (b) (3 points) Find the MGF My (t) when t> . Use MGF to anwser: (c) (2 points) Find Var(Y) (d) (3 points) Find E(Y). 2. (10 points) Let X, Y have joint pdf given by f(x, y) = _ [Ce, 0 < x < y < 2x < 0, O.W. (a) (4 points) Find constant C (b) (3 points) Indentify the marginal distribution of Y (c) (3 points) Indentify the conditional distribution of X given Y = y 3. (10 points) Let X, Y have joint pdf given by 2, 0 y x 1, f(x, y) = 0, o.w. (a) (2 points) Show that whether X and Y are independent or not. (b) (2 points) Find f(x), the marginal density of X. (c) (3 points) Find the conditional pdf f(y|x) and then find E(Y|X) and Var(Y|X). (d) (3 points) Find px,y, the correlation of X and Y. 4. (10 points) An unfair four-sided dice is thrown 4 times. The probability of getting 1s is twice the probability of 2s and half the probability of getting 3s, finally the probability of getting 4s is the same of getting 2s. (a) (3 points) Find the probability that every face appear once. (b) (4 points) Let X be the number of appearances of 2s and Y be the number of appearances of 3s, Z be the number of appearances of the rest. Find the joint distribution of (X,Y) (c) (3 points) Find Cov(X, Y)