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A continuous probability distribution concentrated on the positive semi-axis with density

where is a parameter assuming positive values, and is Euler's gamma-function:

The corresponding distribution function for is zero, and for it is expressed by the formula

The integral on the right-hand side is called the incomplete gamma-function. The density is unimodal and for it attains the maximum at the point . If the density decreases monotonically with increasing , and if , increases without limit. The characteristic function of the gamma-distribution has the form

The moments of the gamma-distribution are given by the formula

In particular, the mathematical expectation and variance are equal to . The set of gamma-distributions is closed with respect to the operation of convolution:

Gamma-distributions play a significant, though not always an explicit, role in applications. In the particular case of one obtains the exponential density. In queueing theory, the gamma-distribution for an which assumes integer values is known as the Erlang distribution. In mathematical statistics gamma-distributions frequently occur owing to the close connection with the normal distribution, since the sum of the squares of independent normally-distributed random variables has density and is known as the "chi-squared" distribution with degrees of freedom. For this reason the gamma-distribution is involved in many important distributions in problems of mathematical statistics dealing with quadratic forms of normally-distributed random variables (e.g. the Student distribution, the Fisher -distribution and the Fisher -distribution). If and are independent and are distributed with densities and , then the random variable has density

which is known as the density of the beta-distribution. The densities of linear functions of random variables obeying the gamma-distribution constitute a special class of distributions — the so-called "type III" family of Pearson distributions. The density of the gamma-distribution is the weight function of the system of orthogonal Laguerre polynomials. The values of the gamma-distribution may be calculated from tables of the incomplete gamma-function [1], [2].


[1] V.I. Pagurova, "Tables of the incomplete gamma-function" , Moscow (1963) (In Russian)
[2] K. Pearson (ed.) , Tables of the incomplete gamma function , Cambridge Univ. Press (1957)



[a1] N.L. Johnson, S. Kotz, "Distributions in statistics" , 1. Continuous univariate distributions , Wiley (1970)
[a2] L.J. Comrie, "Chambers's six-figure mathematical tables" , II , Chambers (1949)
How to Cite This Entry:
Gamma-distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article