09/10 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCESTAT1301 Probability & Statistics IAssignment 2Due Date: October 9, 2009 (Hand in your solutions for Questions 4, 7, 13, 23, 24, 31, 34, 36) 1. Two balls are chosen randomly from an urn containing 6 white, 3 black, and 1 orange balls. Suppose that we win $1 for each white ball drawn and we lose $1 for each orange ball drawn. Denote $Xas the amount that we can win. (a) What are the possible values of X? (b) Determine the probability mass function and cumulative distribution function of X. 2. Two fair dice are rolled. Let Xbe the product of the 2 dice. Determine the pmf of X. 3. A jar contains chips, numbered 1, 2, … nm+nm+. A set of size nis drawn. If we let Xdenote the number of chips drawn having numbers that exceed all of the numbers of those remaining, determine the probability mass function of X. 4. For each of the following, determine the constant csuch that ( )xpsatisfies the conditions of being a pmf for a random variable X, and then compute the mean and variance. (a) , ; ( )cxxp=10,...,3,2,1=x(b) ( )xcxp=, ; 4,3,2,1=x(c) ( )()21+=xcxp, ; 1,0,1−=x(d) , . ( )()!xcxp=3,2,1=x5. Let Xbe a random variable with pmf ( )()912+=xxp, 1,0,1−=xCompute , ()XE()2XEand ()4322+−XXE. 6. A game uses two fair dice. To participate, you pay $20 per roll. You win $10 if even shows, $42 if 7 shows, and $102 if 11 shows. The game is fair if your expected gain is $0. Is the game fair? What is the variance of your gain? 7. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $1.10; if they are different colors, then you win -$1.00 (that is, you lose $1.00). Calculate (a) the expected value of the amount you win; (b) the variance of the amount you win. P. 1
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