11/12 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCESTAT1301 Probability & Statistics IAssignment 5Due Date: December 5, 2011 (Hand in your solutions for Questions 3, 10, 13, 27, 29, 32, 34, 41) 1. Let and Ybe a continuous random variables distributed in (with pdf (1,0~UX))1,0( )yfYand cdf . Suppose Xand Yare independent. Denote as an indicator variable such that if Aoccurs and otherwise. ( )yFY1=AIAI0=AI(a) Show that, for , 10<<ttytyIIytyYYtXP<≥+=⎟⎠⎞⎜⎝⎛=≤|. (b) Using the result in part (a), show that the cdf of XYW=is given by ( )( )( )⎪⎪⎩⎪⎪⎨⎧≥<<+≤=∫1110001ttdyyfyttFttFtYYW. (c) Using variable substitution and integral by parts for the integral in the expression of in part (b), show that ( )tFW( )tFWcan be also expressed as ( )⎪⎪⎩⎪⎪⎨⎧≥<<⎟⎠⎞⎜⎝⎛+≤=∫1110001ttdxxtFtttFtYW. (d) Derive the result in part (c) by first showing that txtxYIIxtFxXXtYP<≥+⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛=≤|for 10<<t. (e) Find the cdf of if the cdf of Yis given by XYW=( )⎪⎩⎪⎨⎧≥<<≤=1110002yyyyyFY. P. 1
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