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stat1301hw5

# stat1301hw5 - 11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 Probability & Statistics I Assignment 5 Due Date: December 5, 2011 (Hand in your solutions for Questions 3, 10, 13, 27, 29, 32, 34, 41) 1. Let and Y be a continuous random variables distributed in ( with pdf ( 1 , 0 ~ U X ) ) 1 , 0 ( ) y f Y and cdf . Suppose X and Y are independent. Denote as an indicator variable such that if A occurs and otherwise. ( ) y F Y 1 = A I A I 0 = A I (a) Show that, for , 1 0 < < t t y t y I I y t y Y Y t X P < + = = | . (b) Using the result in part (a), show that the cdf of XY W = is given by ( ) ( ) ( ) < < + = 1 1 1 0 0 0 1 t t dy y f y t t F t t F t Y Y W . (c) Using variable substitution and integral by parts for the integral in the expression of in part (b), show that ( ) t F W ( ) t F W can be also expressed as ( ) < < + = 1 1 1 0 0 0 1 t t dx x t F t t t F t Y W . (d) Derive the result in part (c) by first showing that t x t x Y I I x t F x X X t Y P < + = = | for 1 0 < < t . (e) Find the cdf of if the cdf of Y is given by XY W = ( ) < < = 1 1 1 0 0 0 2 y y y y y F Y . P. 1

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