places. Problem 1. (17 points) Let X and Y be independent random variables with exponential distributions with rates A = and = , respectively. Find Y-3 P(2Y >3 | X) 2 3 Problem 2. (18 points) Let X = N(3, 3), Y = N(10, 10), U = N(, ), and V = N(, ). Find the probability: 5 3 3. 2 3 20 P(X - Y V - U+1) 5 6 Problem 3. (15 points) Let X be uniformly distributed on (2,5) and Y be uniformly distributed on (1,3). Assume that X and Y are independent random variables. a. What is the joint density of < X,Y >? b. Find the marginal densities of X and Y. c. Find P(2Y 3 | 2Y+X > 6.