Inverse gamma distribution STAT 587 (Engineering) Iowa State - PowerPoint PPT Presentation

Inverse gamma distribution STAT 587 (Engineering) Iowa State University September 17, 2020

Inverse gamma distribution Probability density function Inverse gamma distribution The random variable X has an inverse gamma distribution with shape parameter α > 0 and scale parameter β > 0 if its probability density function is β α Γ( α ) x − α − 1 e − β/x I( x > 0) . f ( x ) = where Γ( α ) is the gamma function, � ∞ x α − 1 e − x dx. Γ( α ) = 0 We write X ∼ IG ( α, β ) .

Inverse gamma distribution Probability density function - graphically Inverse gamma probability density function Inverse gamma random variables scale = 0.5 scale = 1 scale = 2 0.25 0.20 shape = 0.5 0.15 0.10 Probablity density function, f(x) 0.05 0.00 0.3 shape = 1 0.2 0.1 0.0 0.4 shape = 2 0.2 0.0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 x

Inverse gamma distribution Mean and variance Inverse gamma mean and variance If X ∼ IG ( α, β ) , then � ∞ x β α β Γ( α ) x − α − 1 e − β/x dx = · · · = E [ X ] = α − 1 , α > 1 0 and � 2 � ∞ � β β α Γ( α ) x − α − 1 e − β/x dx V ar [ X ] = x − 0 α − 1 β 2 = · · · = α > 2 . ( α − 1) 2 ( α − 2) ,

Inverse gamma distribution Relationship to gamma distribution Relationship to gamma distribution If X ∼ Ga ( α, λ ) where λ is the rate parameter, then Y = 1 X ∼ IG ( α, λ ) .

Inverse gamma distribution Summary Summary Inverse gamma random variable X ∼ IG ( α, β ) , α, β > 0 β α Γ( α ) x − α − 1 e − β/x , x > 0 f ( x ) = β E [ X ] = α − 1 , α > 1 β 2 V ar [ X ] = ( α − 1) 2 ( α − 2) , α > 2