PSTAT W120A, Summer 2016 Assignment 3.A Due Wednesday Aug. 17 Directions: Please write up your solutions and submit them on GauchoSpace by the end of the day on Wednesday. We prefer the submission to be single pdf file if possible. Bonus points will be awarded for organization and neatness. Lesson 9 Joint Distributions 1. Suppose that we have a pair of random variables X and Y with joint distribution Y 2 0.04 0.12 0.24 2 4 6 X 4 0.02 0.06 0.12 6 0.04 0.12 0.24 (a) If we have M = max(X, Y ), the larger of the two observations, then calculate P{M = 4}. (b) Show that X and Y are independent. 2. Suppose that X and Y are independent random variables from binomial experiments where n = 4 in both cases but the probability of success is p = 0.25 for X and p = 0.5 for Y . Calculate the probability of the events (a) {X = 1, Y = 2}, (b) {X + Y = 1}. Lesson 10 Expected Value 3. A random variable X has probability mass function x P (x) -10 0.009 -5 0.064 0 0.198 5 0.316 10 0.272 15 0.120 20 0.021 calculate E(X). 4. I can buy a lottery ticket for $2 which gives me a 1 in 163,000 chance of winning $50,000 and a 1 in 1000 chance of $500 (otherwise you don't win anything.) (a) Calculate the expected profit that I will make if I play. (Profit = Money won - Cost of playing.) (b) If I buy 100 lottery tickets, then calculate the probability that I'll win $50,000 on at least one ticket. (c) Calculate the expected profit from buying 100 lottery tickets. 5. Suppose that X is a random variable that is equally likely to be 1 or -1. Let Y be some other random variable that is independent of X. Prove that h i \u0001 E (X + Y )2 = 1 + E Y 2 . 1 of 1