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# 09_16ans - STAT 410 Examples for Fall 2009 1 2.1.6 Let f x...

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STAT 410 Examples for 09/16/2009 Fall 2009 1. 2.1.6 Let f ( x , y ) = e x y , 0 < x < , 0 < y < , zero elsewhere, be the pdf of X and Y . Then if Z = X + Y , compute P ( Z 0 ) , P ( Z 6 ) , and, more generally, P ( Z z ) , for 0 < z < . What is the pdf of Z ? F Z ( z ) = P ( Z z ) = P ( Y z – X ) = - - - z d x z d y x x y e 0 0 = - - - z d x z d y x x y e e 0 0 = ( 29 + - - - z d x z x x e e 0 1 = - - - z d z z d x x e x e 0 0 = z z e z e - - - - 1 , z > 0. In particular, P ( Z 0 ) = 0, P ( Z 6 ) = 1 – 7 e 6 0.98265. f Z ( z ) = F Z ' ( z ) = e z e z + z e z = z e z , z > 0. Another approach: X and Y are two independent Exponential random variables with mean 1. 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 ) = ( 29 1 1 2 2 1 2 1 z e z - - Γ = z e z , z > 0.

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2. 1. (continued) Let f ( x , y ) = e x y , 0 < x < , 0 < y < , zero elsewhere, be the pdf of X and Y . Determine the probability distribution of the following random variables: b) W = 2 X + Y ; M 2 X + Y ( t ) = E ( e 2 X t + Y t ) = M X ( 2 t ) M Y ( t ) =
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