STAT 230 Final, Fall 2007

American University of Beirut STAT 230 Introduction to Probability and Random Variables Fall 2007 Final Exam Exercise 1 (15 points) Let f(x) = 2 5|x|, if −1 < x < 2, be the pdf of a continuous random variable X. Find the cdf and the pdf of Y = X2. Exercise 2 Let X be a continuous random variable with pdf f(x) = kx2e−x2/2 0 < x < +∞ a. (7 points) find the value of the constant k b. (8 points) find E(X) and V ar(X) Exercise 3 (10 points) Show that probability that the fifth head is observed on the tenth independent flips of a fair coin is 63/512. Exercise 4 (10 points) Roll a pair of fair dice. Let X denote the maximum of the two faces and Y the minimum of the two faces. Compute Cov(X, Y ). Exercise 5 Let X1 and X2 be independent χ2(2) distributions, i.e. f(x) = 1 2 e−x/2 0 < x < +∞ a. (10 points) find the joint pdf of Y1 = X1 + X2 and Y2 = X1 − X2 b. (10 points) find the marginal pdf of Y2 Exercise 6 (15 points) Let X and Y be two independent Binomial random variables with pa- rameters n and p = 1/2. Show that P(X = Y ) = (2n)! (n!)2.22n hint: you may use the identity n � k=0 �n k �2 = �2n n � Exercise 7 (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. (hint: you may use the Central Limit Theorem, and that FZ(−0.29) = 0.39) good luck