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 λ = 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 λ λ λ Procedure to generate a random variable from ( ) λ Exp : [1] Generate ( ) . 1 , 0 ~ U U [2] Compute ( ) U X = 1 log 1 λ . 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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