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STAT1301 (Assig2)

STAT1301 (Assig2) - 09/10 THE UNIVERSITY OF HONG KONG...

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09/10 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability & Statistics I Assignment 2 Due 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 \$ X as 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 X be the product of the 2 dice. Determine the pmf of X . 3. A jar contains chips, numbered 1, 2, … n m + n m + . A set of size n is drawn. If we let X denote 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 c such that ( ) x p satisfies the conditions of being a pmf for a random variable X , and then compute the mean and variance. (a) , ; ( ) cx x p = 10 ,..., 3 , 2 , 1 = x (b) ( ) x c x p = , ; 4 , 3 , 2 , 1 = x (c) ( ) ( ) 2 1 + = x c x p , ; 1 , 0 , 1 = x (d) , . ( ) ( ) ! x c x p = 3 , 2 , 1 = x 5. Let X be a random variable with pmf ( ) ( ) 9 1 2 + = x x p , 1 , 0 , 1 = x Compute , ( ) X E ( ) 2 X E and ( ) 4 3 2 2 + X X E . 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 Two marbles are withdrawn randomly.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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