assignment 2 (ans)

assignment 2 (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: Foundations of Actuarial Science Assignment 2 Solution 1. (a) (b) 2 , 1 , 0 , 1 () = = = = = 2 3 1 1 5 2 0 5 1 1 15 1 x x x x x p (c) < < < < = 2 1 2 1 3 2 1 0 15 4 0 1 15 1 1 0 x x x x x x F 2. = = = = = 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 3. + + = n n m i n i n m i p 1 , n i ,..., 2 , 1 , 0 = 4. (a) 55 1 = c , 7 = μ , (b) 6 2 = σ 25 12 = c , 25 48 = , 625 696 2 = (c) 9 1 = c , 0 = , 9 8 2 = (d) 9 1 = c , 9 23 = , 81 38 2 = 5. 0, 9 8 , 9 52 6. Not fair, 9 5489 7. Let X be the gain in the game. Then ( ) 9 4 2 9 4 10 5 color same ball both 1 . 1 = × × = = = P X P 9 5 9 4 1 1 = = = X P (a) ()( ) 15 1 9 5 1 9 4 1 . 1 = + = X E (b) ()() 75 82 9 5 1 9 4 1 . 1 2 2 2 = + = X E () () () 45 49 225 1 75 82 2 2 = = = X E X E X Var
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10/11 p. 2 8. (a) 29 (b) 61.72 (c) Both are telling the truth. 9. () 2 1 1 0 + + = N N X E , ( ) 12 1 2 0 1 = N N X Var 10. A p 1 . 0 + 11. (a) 14 (b) 45 (c) 9 4 12. (a) 3.88% (b) 53.19% (c) 2.06% 13. (a) = otherwise 0 if 1 1 N i n n N n i i p (b) ( ) 1 1 + + n N n , ( ) 2 2 1 + + + n N N n 14. 3 5 = X E , 36 41 = X Var , 2 1 1 = p , 3 1 2 3 = p , 6 1 4 = p 15. (a) p p e t M t X = 1 log 1 log , p t log < (b) p p p 1 log 1 , ()() + p p p p p 1 log 1 1 log 1 2 18. (a) 4 3 (b) < < + = 1 1 1 1 3 2 4 1 1 0 3 x x x x x x F X (c) 0, 5 1 (d) 0.671 19. (a) > = 10 0 10 10 1 x x x x F (b) 3 2 (c) no finite mean and variance (d) 6.3246 20. (a) = y t Y dt e y F 2 1 λ For , 0 y y y t y t Y e e dt e y F 2 1 2 1 2 1 = = = For , 0 > y () () + = y t Y Y dt e F y F 0 2 1 0 y y t e e = + = 2 1 1 2 1 2 1 0
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10/11 p. 3 Hence () < < < = y e y e y F y y Y 0 if 2 1 1 0 if 2 1 .
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assignment 2 (ans) - 10/11 THE UNIVERSITY OF HONG KONG...

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