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stat1301hw3sol

stat1301hw3sol - 11/12 THE UNIVERSITY OF HONG KONG...

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11/12 p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability and Statistics I Assignment 3 Solution 1. (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) 0.6575 (e) ( ) = y y y f Y 1 4 3 , 1 0 < < y 2. (a) For , 5 > x ( ) ( ) x t dt t x X P x F x x 5 1 5 5 5 5 2 = = = = Hence ( ) > = 5 0 5 5 1 x x x x F . (b) ( ) ( ) 12 5 12 5 1 1 12 1 12 = = = F X P (c) ( ) [ ] = = = × = 5 5 5 2 log 5 5 5 x dx x dx x x X E ( ) [ ] = = = × = 5 5 5 2 2 2 5 5 5 dx dx x x X E The random variable X does not have finite mean and finite variance. (d) ( ) [ ] 472 . 4 5 10 10 5 5 5 2 1 5 2 3 5 2 = = = = × = x dx x dx x x X E (e) ( ) ( ) 12 5 12 hours 12 least at for function can device a = = X P P Let Y be the number of devices out of 8 that can function for at least 12 hours. Then with the assumption of independence among the lifetimes, 12 5 , 8 ~ b Y . Hence ( ) 7185 . 0 12 7 12 5 2 8 12 7 12 5 1 8 12 7 1 3 6 2 7 8 = = Y P .
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