Gamma Distribution
The gamma distribution is a non-negative continuous probability distribution function.
Contents
Description
The distribution is characterized by shape and scale parameters, both of which are strictly positive. A random variable distributed by this function is expressed as X ~ Gamma(α, β).
The distribution is given as:
for any x ∈ [0,,∞), and where Γ is the gamma function.
In some cases, the distribution is parameterized to a rate (θ) instead of scale (β). Note that rate is the inverse of scale, i.e. θ = 1/β, so the substitution is very simple to apply. The distribution becomes:
The exponential distribution is equivalent to a gamma distribution parameterized for rate with shape fixed at 1: Gamma(1, θ) = Exp(θ).
Moments
The expected value is given as E[X] = αβ = α/θ.
Variance is given as Var[X] = αβ2 = α/(θ2).