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# 10_15 - STAT 410 Examples for Fall 2008 Gamma Distribution...

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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 = a Y + b , then M W ( t ) = e b t M Y ( a t ) . M 2 Y / θ ( t ) = M Y ( 2 t / θ ) = ( ) ± 2 1 1 t - .

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