Gamma distribution

The gamma distribution is a real continuous Probability distribution defined by a shape parameters α\alpha and a scale parameter τ\tau. For a Random variable XX, the Probability density function is

fX(x;α,τ)=1Γ(α)1ταxα−1e−x/τf_{X}(x;\alpha,\tau)=\frac{1}{\Gamma(\alpha)} \frac{1}{\tau^{\alpha}}x^{\alpha-1}e^{-x/\tau}

where Γ\Gamma is the Gamma function. Both α\alpha and τ\tau are positive real numbers. Equivalently, it can be parameterized as

f(x;α,λ)=1Γ(α)λαxα−1e−λxf(x;\alpha,\lambda)=\frac{1}{\Gamma(\alpha)}\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}

where λ\lambda is a positive real rate parameter. The two are related by λ=1/τ\lambda=1/\tau.

It is a very general distribution that has applications in many fields, such as modeling wait times in queue theory, polymer chemistry and more. It is most often used with specific parameters to form more specific distributions; see > Relation to other distributions below.

Moments

The central and raw moment-generating function for the Gaussian are

MX∗(t)=(1−τt)−α,MX(t)=e−tατ(1−τt)−αM_{X}^{*}(t)=(1-\tau t)^{-\alpha},\qquad M_{X}(t)=e^{-t\alpha \tau}(1-\tau t)^{-\alpha}

The expected value is

E[X]=ατE[X]=\alpha \tau

and the variance is

var(X)=ατ2\text{var}(X)=\alpha \tau ^{2}

Relation to other distributions

Specific cases of this distribution are themselves well-known and more often used.

  • For α=1\alpha=1 and any τ\tau, we get an Exponential distribution, Exp(x;τ)\text{Exp}(x;\tau).1
  • For integer α\alpha and any τ\tau, we get an Erlang distribution, Erlang(x;α,τ)\text{Erlang}(x;\alpha,\tau).
  • For positive integer α\alpha and τ=2\tau=2, we get a Chi-square distribution with 2α2\alpha degrees of freedom, χ2α2(x)\chi ^{2}_{2\alpha}(x).

Footnotes

  1. Take care with the parameterization. A "scale parameter" Γ\Gamma distribution maps to a "scale parameter" exponential or Erlang. Similarly, rate parameter maps to rate parameter. ↩