Gamma Distribution

(continuous probability dist for equations)

Usage:

GammaDist (x, α, β)

Definition:

 x^(α-1) exp(-x/ β) / (gamma(α) β ^ α)

 = exp[(α-1)log(x) - x/ β - log(gamma(α)) - α log(β)]

Parameters:

α is shape     β is scale

Required:

α > 0          β > 0

Support:

x ⋝ 0

Moments:

μ = α β               σ = β sqrt (α)

γ1 = 2 / sqrt (α)     β2 = 3 + 6 / α

If events occur by a Poisson process, then the time required for the occurrence of α events is described by the gamma distribution (where β is the average time between events).

For α = 1, this is the exponential distribution with λ = 1 / β. For β = 2, this is the chi-square distribution with degrees of freedom ν = 2α.

The Erlang distribution is a special case of the gamma distribution in which β = 1 and α = n (which is an integer).