STAT 230 Final, Summer 2008

American University of Beirut STAT 230 Introduction to Prob0ability and Random Variables Summer 2008 Final Exam Name: ............................... ID #: ................................. Section: 4 (7:30) 6 (12:30) Exercise 1 (10 points) The number of accidents in a period of one week follows a Poisson distribution with mean two. The number of accidents from week to week are independent. Find the probability of exactly seven accidents in a given three weeks. Exercise 2 Let X be a continuous random variable with pdf f(x) = kx2e−x2/2 , 0 < x < +∞ a. (10 points) find the value of the constant k b. (10 points) find E(X) and V ar(X) Exercise 3 (20 points) Let X1 and X2 be independent χ2(2) distributions, i.e. f(x) = 1 2 e−x/2 0 < x < +∞ find the joint pdf of Y1 = X1 + X2 and Y2 = X1 − X2, then find the marginal pdf of Y2 Exercise 4 (20 points) Flip n = 8 fair coins and remove all that come up heads. Flip the other (tails) coins and remove the heads. Continue flipping the remaining coins until each has come up heads. We shall find the pdf of Y , the number of trials needed. Let Xi, equal the number of flips required to observe heads on coin i, i = 1, 2, . . . , 8. Then Y = max(X1, X2, . . . , X8). a. Show that P(Y ≤ y) = [1 − (1/2)y]8 b. Show that P(Y = y) = [1 − (1/2)y]8 − [1 − (1/2)y−1]8 , y = 1, 2, .. c. Find E(Y ) Exercise 5 (15 points) Suppose that the length of life in hours of a light bulb manufactured by company A is N(800, 14400) and the length of life in hours of a light bulb manufactured by company B is N(850, 2500). One bulb is selected from each company and is burned until death. a. find the probability that length life of the bulb from company A exceeds the length of life of the bulb from company B by at least 15 hours. b. find the probability that at least one of the bulbs lives for at least 920 hours. Exercise 6 (15 points) On each bet, a gambler loses 1 with probability 0.7, loses 2 with prob- ability 0.2, or wins 10 with probability 0.1. Approximate the probability that the gambler will be losing after the first 100 bets.