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 f(x) = βα Γ(α)x−α−1e−β/x I(x > 0). where Γ(α) is the gamma function, Γ(α) = ∞ xα−1e−xdx. We write X ∼ IG(α, β).

Inverse gamma distribution Probability density function - graphically

Inverse gamma probability density function

scale = 0.5 scale = 1 scale = 2 shape = 0.5 shape = 1 shape = 2 1 2 3 4 1 2 3 4 1 2 3 4 0.00 0.05 0.10 0.15 0.20 0.25 0.0 0.1 0.2 0.3 0.0 0.2 0.4

x Probablity density function, f(x)

Inverse gamma random variables

Inverse gamma distribution Mean and variance

Inverse gamma mean and variance

If X ∼ IG(α, β), then E[X] = ∞ x βα Γ(α)x−α−1e−β/xdx = · · · = β α − 1, α > 1 and V ar[X] = ∞

• x −

β α−1

2

βα Γ(α)x−α−1e−β/xdx

= · · · =

β2 (α−1)2(α−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 f(x) =

βα Γ(α)x−α−1e−β/x, x > 0

E[X] =

β α−1, α > 1

V ar[X] =

β2 (α−1)2(α−2), α > 2