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Unformatted text preview: 11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability & Statistics I Assignment 2 Due Date: October 14, 2011 (Hand in your solutions for Questions 2, 17, 18, 19, 25, 27, 36, 37) 1. The probability mass distribution for a discrete random variable X is as shown in the following table: X 0 1 2 3 4 5 ( ) x p 0.05 0.3 ? 0.2 0.1 0.05 (a) What is the value of ? ( ) 2 p (b) Find the values of μ and . 2 σ (c) Find the probability that X falls in the interval σ μ 2 ± . 2. The maximum patent life for a new drug is 17 years. Subtracting the length of time required for testing and approval of the drug by the Food and Drug Administration, you obtain the actual patent life of the drug – that is, the length of time that a company has to recover research and development costs and make a profit. Suppose that the distribution of the length of patent life for new drugs is as shown in the following table: Years ( X ) 3 4 5 6 7 8 9 10 11 12 13 ( ) x p 0.03 0.05 0.07 0.10 0.14 0.20 0.18 0.12 0.07 0.03 0.01 (a) Find the expected number of years of patent life for a new drug. (b) Find the standard deviation of X . (c) Find the probability that X falls within two standard deviations from the mean. (d) Suppose the cost for acquiring the patent for a new drug is $13000 and the cost for maintaining the patent is $2000 for each year. Find the expected value and the standard deviation of the cost for patenting a new drug. 3. Two balls are chosen randomly from an urn containing 7 white, 4 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 . 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 = 8 ,..., 3 , 2 , 1 = x (b) ( ) x c x p = , ; 5 , 4 , 3 , 2 , 1 = x (c) ( ) ( ) 2 1 + = x c x p , ; 2 , , 2 − = x (d) , . ( ) ( ) ! x c x p = 4 , 3 , 2 , 1 = x P. 1 11/12 5. Two fair dice are rolled. Let X be the product of the 2 dice. Determine the pmf of X . 6. 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 . 7. Let X be a random variable with pmf ( ) ( ) 9 1 2 + = x x p , 1 , , 1 − = x Compute , ( ) X E ( ) 2 X E and ( ) 3 2 4 2 + − X X E . 8. A game uses two fair dice. To participate, you pay $20 per roll. You get $8 if even shows, $55 if 7 shows, and $123 if 11 shows. The game is fair if your expected gain is $0. Is the game fair? What is the variance of your gain?...
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