assignment 2

# assignment 2 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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Unformatted text preview: 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 Probability and Statistics: Foundations of Actuarial Science Assignment 2 Due Date: October 11, 2010 (Hand in your solutions for Questions 7, 20, 21, 25, 26, 29, 40, 43) 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 , , 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 , , 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 Two marbles are withdrawn randomly.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 10/11 8. In a large New England college, there are 20 classes of introductory statistics with the following distribution of class size. Class Size Relative Frequency 10 0.5 20 0.3 90 0.2 Total 1 The student newspaper reported that the average statistics student faced a class size “over 50.” Alarmed, the Dean asked the 20 professors to calculate their average class sizes, and they reported, “under 30.” (a) What class size does the average professor have? (b) What class size does the average student have? (c) Who is telling the truth? 9. If X takes on one of the number in the set { 1 + N , 2 + N , …, } ( ) with equal probabilities, i.e. 1 N 1 N N < ( ) 1 1 N N i X P − = = , 1 ,..., 2 , 1 N N N i + + = ; then X is said to have an integer uniform distribution, denoted by . Find the mean and variance of X ....
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assignment 2 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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