assignment 3 (ans)

assignment 3 (ans) - 10/11 THE UNIVERSITY OF HONG KONG...

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10/11 p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 Probability and Statistics: Foundation of Actuarial Science Assignment 3 Solution 1. (a) 3 2 (b) 3 2 2. (a) (b) () 1 , a U a U , 0 3. (a) From the definition of , we know that 1 F ( ) ( ) u u F F 1 for all . Since F is a non-decreasing function, we have ( 1 , 0 u ) () () ( ) ( ) ( ) x F u x F u F F x u F 1 1 (as ( ) u u F F 1 ) On the other hand, since ( ) u F 1 is the minimum x such that , therefore if then u x F u x F x must be greater than or equal to ( ) u F 1 , i.e. ( ) ( ) u F x F 1 x u . (b) The distribution function of X is given by () ( ) ( ) ( ) ( ) ( ) ( ) x F x F U P x U F P x X P x F X = = = = 1 . (c) Distribution function of λ Exp is ( ) x e x F = 1, . Therefore 0 > x {} u u x x u e x u F x x x = = = 1 log 1 1 log 1 : min 1 : min 1 Procedure to generate a random variable from ( ) Exp : [1] Generate . 1 , 0 ~ U U [2] Compute U X = 1 log 1 . 4. 97 5. 0.1719 6. 5 . 0 > p 7. Let X be the number of times the stock increases in n weeks. Then . Denote $ Y as the price of the stock after n weeks, then ( 55 . 0 , ~ n b X ) ( ) X n X n X n X Y 5625 . 1 8 . 0 10 8 . 0 25 . 1 8 . 0 10 8 . 0 25 . 1 10 × = = × = .
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10/11 p. 2 (a) For , 4 = n () ( ) ( ) ( ) 3910 . 0 45 . 0 55 . 0 4 4 45 . 0 55 . 0 3 4 3 9085 . 2 15 5625 . 1 8 . 0 10 15 0 4 1 3 4 = + = = > = > × = > X P X P P Y P X (b) () ( ) ( ) ( ) X n E Y E 5625 . 1 8 . 0 10 × = 5625 . 1 log 8 . 0 10 X n e E × = () ( 5625 . 1 log 8 . 0 10 X n M × = ) n n e 55 . 0 1 55 . 0 8 . 0 10 5625 . 1 log + × × × = n 0475 . 1 10 × = For , 2 = n ( )() 9726 . 1 0475 . 1 10 2 = × = Y E For , 4 = n ( 0397 . 12 0475 . 1 10 4 = × = Y E 8. 3 9. (a) (i) (ii) (iii) ( 3 1 p ) ) ( 2 1 3 p p ( ) p p 1 3 2 (iv) 3 p (b) (i) 0 (ii) (iii) ( 2 1 p ) ( ) p p 1 2 (iv) 2 p (c) p 10. (a) 0.3679 (b) 0.9412 11. , () () ( ) ( ) [ 2 3 2 1 1 1 30 ' t t T T e t e t t t F t f + + = = ] 0 > t 13. p p p r r 1 1 14. 0.3697 16. 5 3 17. (a) Let X be the number of Aces in the hand dealt to the North. Then X is a Hypergeometric random variable with , 52 = N 4 = m , 13 = n . The probability mass function of X is given by () ( ) 4 , 3 , 2 , 1 , 0 , 13 52 13 48 4 = =
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assignment 3 (ans) - 10/11 THE UNIVERSITY OF HONG KONG...

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