09_24ans - STAT 410 Examples for 09/24/2008 Fall 2008 1....

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Unformatted text preview: STAT 410 Examples for 09/24/2008 Fall 2008 1. Let X and Y be two independent Exponential random variables with mean 1. Find the probability distribution of Z = X + Y. That is, find ( ) z f Z = ( ) z f Y X + . Recall 2.1.6 ( Homework 3 ): 2.1.6 Let f ( x , y ) = e x y , 0 < x < , 0 < y < , zero elsewhere, be the p.d.f. of X and Y. Then if Z = X + Y, What is the p.d.f. of Z? F Z ( z ) = P ( Z z ) = P ( Y z X ) = & & --- z d x z d y x x y e = & & --- z d x z d y x x y e e = ( ) & +--- z d x z x x e e 1 = & &--- z d z z d x x e x e = z z e z e---- 1 , z > 0. f Z ( z ) = F Z ' ( z ) = e z e z + z e z = z e z , z > 0. Another approach: M X ( t ) = t- 1 1 , t < 1. M Y ( t ) = t- 1 1 , t < 1. M X + Y ( t ) = M X ( t ) M Y ( t ) = 2 1 1 - t , t < 1. Z = X + Y has a Gamma distribution with = 2, = 1. f Z ( z ) = ( ) 1 1 2 2 1 2 1 z e z-- = z e z , z > 0. 2. Let X and Y be two independent Exponential random variables with mean 1. Find the p.d.f. of Z = 2 X + Y. M 2 X + Y ( t ) = E ( e 2 X t + Y t ) = M X ( 2 t ) M Y ( t ) = & - & - t t 1 1 2 1 1 , t < 1 / 2 . f 2 X + Y ( z ) = ???...
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09_24ans - STAT 410 Examples for 09/24/2008 Fall 2008 1....

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