stat1301_assignment03_ans

stat1301_assignment03_ans - 09/10 THE UNIVERSITY OF HONG...

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09/10 p. 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability and Statistics I Assignment 3 Solution 1. (a) () 3 4 3 1 1 1 3 1 1 2 = = x x dx x Hence 4 3 = c . (b) For , 1 1 < < x + = = = 3 2 3 4 3 3 4 3 1 4 3 3 1 3 1 2 x x t t dt t x F x x X ( ) 3 3 2 4 1 x x + = Hence < < + = 1 1 1 1 3 2 4 1 1 0 3 x x x x x x F X . (c) 0 4 3 1 1 3 = = dx x x X E (the integrant is an odd function) 5 1 5 3 4 3 4 3 1 1 5 3 1 1 4 2 2 = = = x x dx x x X E () () () 5 1 2 2 = = X E X E X Var (d) ( ) ( ) 3 . 0 7 . 0 7 . 0 3 . 0 = < F F X P ( ) 3 3 3 . 0 3 . 0 3 2 4 1 7 . 0 7 . 0 3 2 4 1 × + × + = 671 . 0 = (e) For , 1 0 < < y () ( ) ( ) y X y P y X P y Y P y F Y = = = 2 ( ) ( ) y F y F X X = 2 3 2 3 3 2 4 1 3 2 4 1 y y y y + + = ( 2 3 3 2 1 y y = ) Hence () () = = = y y y y y F y f Y Y 1 4 3 4 3 4 3 ' 2 1 2 1 , . 1 0 < < y
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09/10 p. 2 2. (a) () > = 10 0 10 10 1 x x x x F (b) 3 2 (c) no finite mean and variance (d) 6.3246 (e) 3 40 , 20, 40 (f) 0.8999 3. (a) From the definition of F , we know that ( ) ( ) u u F F for all ( ) 1 , 0 u . Since F is a non- decreasing function, we have ( ) ( ) x F u x F u F F x u F ( ( ) ( ) u u F F )
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stat1301_assignment03_ans - 09/10 THE UNIVERSITY OF HONG...

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