# Difference between revisions of "Gamma-distribution"

Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
|||

Line 23: | Line 23: | ||

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330021.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330021.png" /></td> </tr></table> | ||

− | Gamma-distributions play a significant, though not always an explicit, role in applications. In the particular case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330022.png" /> one obtains the exponential density. In queueing theory, the gamma-distribution for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330023.png" /> which assumes integer values is known as the [[Erlang distribution|Erlang distribution]]. In mathematical statistics gamma-distributions frequently occur owing to the close connection with the normal distribution, since the sum of the squares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330024.png" /> of independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330025.png" /> normally-distributed random variables has density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330026.png" /> and is known as the "chi-squared" distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330027.png" /> 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|Student distribution]], the [[Fisher-F-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330028.png" />-distribution]] and the [[Fisher | + | Gamma-distributions play a significant, though not always an explicit, role in applications. In the particular case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330022.png" /> one obtains the exponential density. In queueing theory, the gamma-distribution for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330023.png" /> which assumes integer values is known as the [[Erlang distribution|Erlang distribution]]. In mathematical statistics gamma-distributions frequently occur owing to the close connection with the normal distribution, since the sum of the squares <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330024.png" /> of independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330025.png" /> normally-distributed random variables has density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330026.png" /> and is known as the "chi-squared" distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330027.png" /> 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|Student distribution]], the [[Fisher-F-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330028.png" />-distribution]] and the [[Fisher z-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330029.png" />-distribution]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330031.png" /> are independent and are distributed with densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330033.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330034.png" /> has density |

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330035.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330035.png" /></td> </tr></table> |

## Latest revision as of 10:27, 19 October 2014

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].

#### References

[1] | V.I. Pagurova, "Tables of the incomplete gamma-function" , Moscow (1963) (In Russian) MR0159040 |

[2] | K. Pearson (ed.) , Tables of the incomplete gamma function , Cambridge Univ. Press (1957) |

#### Comments

#### References

[a1] | N.L. Johnson, S. Kotz, "Distributions in statistics" , 1. Continuous univariate distributions , Wiley (1970) MR0270476 MR0270475 Zbl 0213.21101 |

[a2] | L.J. Comrie, "Chambers's six-figure mathematical tables" , II , Chambers (1949) |

**How to Cite This Entry:**

Gamma-distribution.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Gamma-distribution&oldid=24074