Gamma distribution - Definition and Overview

In probability theory and statistics, the gamma distribution is a continuous probability distribution.

Specification of the gamma distribution

Probability density function

The probability density function of the gamma distribution can be expressed in terms of the gamma function:

<math> f(x) = x^{k-1} \frac{e^{-x/\theta}}{\Gamma(k)\,\theta^k}

\ \ \ \ \mathrm{for\ } x > 0<math>

where k > 0 is the shape parameter and θ > 0 is the scale parameter of the gamma distribution.

Alternatively, the gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ:

<math> g(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; x^{\alpha-1} \; \exp(-\beta\,x) <math>

Cumulative distribution function

The cumulative distribution function can be expressed in terms of the incomplete gamma function,

<math> F(x) = \int_0^x f(u)\,du

 = \frac{\gamma(k, x/\theta)}{\Gamma(k)} <math>

Properties

Moments

Let X be a random variable following a gamma distribution with parameters k and θ. Then X has the following properties:

mode 

<math>(k-1)\,\theta<math> for k ≥ 1

mean 

<math>\mu = k\,\theta<math>

variance 

<math>\sigma^2 = k\,\theta^2<math>

skewness 

<math>\gamma_1 = \frac{2}{\sqrt{k}}<math>

kurtosis 

<math>\gamma_2 = \frac{6}{k}<math>

Relation to other distributions

If X₁ has a gamma distribution with parameters k₁ and θ, and X₂ has a gamma distribution with parameters k₂ and θ, then X₁ + X₂ has a gamma distribution with parameters k₁ + k₂ and θ.

If k is equal to 1, the gamma distribution is an exponential distribution with parameter θ. The sum of n independent exponential random variables, all with the same parameter θ, is a gamma variable with parameters n and θ.

If k is an integer, the gamma distribution is an Erlang distribution (so named in honor of A. K. Erlang) and is the probability distribution of the waiting time of the kth "arrival" in a one-dimensional Poisson process with intensity 1/θ.

If k is a half-integer and θ = 2, then the gamma distribution is a chi-square distribution with 2 k degrees of freedom.

The gamma distributions are infinitely divisible probability distributions.

References

• R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)