stat2601hw2sol.pdf - 12/13 Spring THE UNIVERSITY OF HONG...

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12/13 Spring p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT2601 Probability and Statistics I Assignment 2 Solution 1. (a) 0.3 (b) 6.15, 1.5275 (c) 0.95 2. (a) ( ) 9 . 7 01 . 0 13 05 . 0 4 03 . 0 3 = × + + × + × = L X E The expected number of years of patent life for a new drug is 7.9 years. (b) ( ) 14 . 67 01 . 0 13 05 . 0 4 03 . 0 3 2 2 2 2 = × + + × + × = L X E ( ) ( ) 73 . 4 9 . 7 14 . 67 2 = = X Var The standard deviation of X is 17 . 2 73 . 4 = years. (c) 56 . 3 17 . 2 2 9 . 7 2 = × = σ μ 24 . 12 17 . 2 2 9 . 7 2 = × + = + σ μ ( ) ( ) ( ) ( ) 12 5 4 2 2 = + + = + = = + < < X P X P X P X P L σ μ σ μ 96 . 0 01 . 0 03 . 0 1 = = (d) Let Y be the cost for patenting a new drug. Then X Y 2000 13000 + = . ( ) ( ) 28800 9 . 7 2000 13000 2000 13000 = × + = + = X E Y E ( ) ( ) 18920000 73 . 4 4000000 2000 2 = × = = X Var Y Var 4349 18920000 = = Y σ The expected cost is $28800, with a standard deviation of $4349. 3. (a) { –1,0, 1, 2 } (b) ( ) = = = = = 2 22 7 1 33 14 0 66 13 1 33 2 x x x x x p (c) ( ) < < < < = 2 if 1 2 1 if 33 14 1 0 if 66 13 0 1 if 33 2 1 if 0 x x x x x x F 4. (a) 36 1 , 3 17 , 9 35 (b) 137 60 , 137 300 , 18769 33300 (c) 19 1 , 0, 19 72 (d) 33 1 , 33 11 9 , 1089 590 5. ( ) = = = = = 12 , 6 9 1 4 12 1 30 , 24 , 20 , 18 , 15 , 10 , 8 , 5 , 3 , 2 18 1 36 , 25 , 16 , 9 , 1 36 1 i i i i i p . ( ) + + = n n m x n x n m x p 1 , n x ,..., 2 , 1 , 0 = . 6
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12/13 Spring p. 2 0, 9 8 , 9 59 7. 8. Fair, 3 2930 ) 5 6 9. (a) 0 (b 10. (a) 25 (b) 3 5 .2 (c) Both 11. 2 1 1 0 + + N N , ( ) 12 1 2 0 1 N N 3. ( ) A p 05 . 0 + 12. ( ) ( ) ( ) 27 2 2 = + = X E X Var X E 1 (a) ( ) [ ] ( ) ( ) 66 6 9 3 2 2 = + + = + X E X E X E ( ) ( ) 32 16 4 3 = × = X Var X Var (b) (c) By Chebyshev’s inequality, ( ) ( ) ( ) 18 17 2 6 1 1 2 6 6 5 2 = = σ μ X P . 4. (a) 2.8%, 3.75% (b) 76.92% (c) 2.88% 5. (a) 11 1 = X P X P 1 ( ) if 1 N x n N y ( ) 1 1 + + n N n =
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