Gamma distribution STAT 587 (Engineering) Iowa State University - - PowerPoint PPT Presentation

Gamma distribution

STAT 587 (Engineering) Iowa State University

September 17, 2020

Gamma distribution Probability density function

Gamma distribution

The random variable X has a gamma distribution with shape parameter α > 0 and rate parameter λ > 0 if its probability density function is p(x|α, λ) = λα Γ(α)xα−1e−λx I(x > 0) where Γ(α) is the gamma function, Γ(α) = ∞ xα−1e−xdx. We write X ∼ Ga(α, λ).

Gamma distribution Probability density function - graphically

Gamma probability density function

rate = 0.5 rate = 1 rate = 2 shape = 0.5 shape = 1 shape = 2 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6

x Probablity density function, f(x)

Gamma random variables

Gamma distribution Mean and variance

Gamma mean and variance

If X ∼ Ga(α, λ), then E[X] = ∞ x λα Γ(α)xα−1e−λxdx = · · · = α λ and V ar[X] = ∞

• x − α

λ 2 λα Γ(α)xα−1e−λxdx = · · · = α λ2 .

Gamma distribution Cumulative distribution function

Gamma cumulative distribution function

If X ∼ Ga(α, λ), then its cumulative distribution function is F(x) = x λα Γ(α)tα−1e−λtdt = · · · = γ(α, βx) Γ(α) where γ(α, βx) is the incomplete gamma function, i.e. γ(α, βx) = βx tα−1e−tdt.

Gamma distribution Cumulative distribution function - graphically

Gamma cumulative distribution function - graphically

rate = 0.5 rate = 1 rate = 2 shape = 0.5 shape = 1 shape = 2 1 2 3 4 1 2 3 4 1 2 3 4 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00

x Cumulative distribution function, F(x)

Gamma random variables

Gamma distribution Relationship to exponential distribution

Relationship to exponential distribution

If Xi

iid

∼ Exp(λ), then Y =

n

• i=1

Xi ∼ Ga(n, λ). Thus, Ga(1, λ) d = Exp(λ).

Gamma distribution Parameterization by the scale

Parameterization by the scale

A common alternative parameterization of the Gamma distribution uses the scale θ = 1

λ. In

this parameterization, we have f(x) = 1 Γ(α)θα xα−1e−x/θ I(x > 0) and E[X] = αθ and V ar[X] = αθ2.

Gamma distribution Summary

Summary

Gamma random variable X ∼ Ga(α, λ), α, λ > 0 f(x) =

λα Γ(α)xα−1e−λx, x > 0

E[X] = α

λ

V ar[X] = α

λ2