# 09_24ans - STAT 410 Examples for Fall 2008 1 Let X and Y be...

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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 α = 2, θ = 1. § f Z ( z ) = ( ) 1 1 2 2 1 2 1 z e z-- ⋅ ⋅ ⋅ Γ = 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 Fall 2008 1 Let X and Y be...

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