Gamma distribution

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Gamma distribution

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The Gamma distribution is a two-parameter family of functions (optionally three parameter family) that is a generalization of the Exponential distribution and closely related to many other forms of continuous distribution. The most general form of the probability density function is:

where

The function is also written with the parameter β as its inverse θ=1/β, giving the form:

The two-parameter form of the distribution is obtained by letting γ=0:

The standard form of the Gamma is formed by setting γ=0, and β=1:

If α=1 this distribution simplifies further an becomes the Exponential distribution (see graph below, first curve). If α is an integer this distribution becomes an Erlang distribution. If α=ν/2 and β=2 the standard Gamma is known as a Chi-square distribution, generally written in the form:

The sum of two or more Gamma distributed random variables is a Gamma variable, and the ratio of a Gamma variable to the sum of two Gamma variables yields a variable that is distributed as a Beta. Johnson and Kotz (1970, [JOH1]) provide details of a wide range of variants of the Gamma distribution, including: truncation, e.g. truncation from above, common in lifetime-testing, where there is some fixed time limit imposed; compounding, where a Gamma distribution is modified by treating one (or more) parameters as itself being distributed according to a standard distribution, such as another Gamma or a Poisson; transformed Gamma distributions, whereby a function such as the log, square root, or cube root of a Gamma (or Chi-square) variate is taken, typically to yield an approximately Normal variate; and generalized Gamma distributions, which involve cases such as those discussed earlier, whereby we consider a variable such as Y=(X-θ)c that has an Exponential distribution, noting that X then has a Weibull distribution.

Plots of the Gamma distribution (single parameter version) are shown below. As can be seen, the distribution form ranges from the Exponential (sometimes called the Negative Exponential) to a form that becomes approximately Normal, but has the 'advantage' that it is only defined for positive values of x.

Gamma distribution curves: α =1,2,...10, β=1

gammadis

Key measures for the distribution are provided in the table below (α,β>0) :

Mean, μ1

αβ

Mode

(α-1)β, α>1

Variance, μ2

αβ2

Skewness

2/√α

Kurtosis

6/α

MGF

References

[JOH1] Johnson N L, Kotz S (1970) Continuous Univariate Distributions, I. Houghton Mifflin/J Wiley & Sons, New York

Mathworld: Weisstein E W: Gamma distribution: http://mathworld.wolfram.com/GammaDistribution.html

Wikipedia: Gamma distribution: http://en.wikipedia.org/wiki/Gamma_distribution