10_15 - STAT 410 Examples for 10/15/2008 Fall 2008 Gamma...

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STAT 410 Examples for 10/15/2008 Fall 2008 Gamma Distribution : ( ) ( ) x e x x f 1 ± ± ± & - - Γ = , 0 x < OR ( ) ( ) ² 1 ± 1 ± ² ± x e x x f - - Γ = , 0 x < If T has a Gamma ( α , θ = 1 / λ ) distribution, where α is an integer, then F T ( t ) = P ( T t ) = P ( X t α ) , where X t has a Poisson ( λ t ) distribution. 1. a) Let X be a random variable with a chi-square distribution with r degrees of freedom. Show that X has a Gamma distribution. What are α and θ ? M X ( t ) = M χ 2 ( r ) ( t ) = ( ) 2 2 1 1 r t - , t < 2. M Gamma ( α , θ ) ( t ) = ( ) ± 1 1 ² t - , t < θ . If X has a chi-square distribution with r degrees of freedom, then X has a Gamma distribution with α = r / 2 and θ = 2. b) Let Y be a random variable with a Gamma distribution with parameters α and θ = 1 / λ . Assume α is an integer. Show that 2 Y / θ has a chi-square distribution. What is the number of degrees of freedom? M Y ( t ) = M Gamma ( α , θ ) ( t ) = ( ) ± 1 1 ² t - , t < θ . If W =
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This note was uploaded on 10/26/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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10_15 - STAT 410 Examples for 10/15/2008 Fall 2008 Gamma...

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