{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 - .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}