View source for Gamma-distribution

A continuous probability distribution concentrated on the positive semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g0433001.png" /> with 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/g0433002.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g0433003.png" /> is a parameter assuming positive values, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g0433004.png" /> is Euler's gamma-function:

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The corresponding distribution function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g0433006.png" /> is zero, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g0433007.png" /> it is expressed by the formula

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The integral on the right-hand side is called the incomplete gamma-function. The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g0433009.png" /> is unimodal and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330010.png" /> it attains the maximum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330011.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330013.png" /> the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330014.png" /> decreases monotonically with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330015.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330017.png" /> increases without limit. The characteristic function of the gamma-distribution has the form

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

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In particular, the mathematical expectation and variance are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330020.png" />. The set of gamma-distributions is closed with respect to the operation of convolution:

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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

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which is known as the density of the beta-distribution. The densities of linear functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330036.png" /> of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043300/g04330037.png" /> 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|Laguerre polynomials]]. The values of the gamma-distribution may be calculated from tables of the incomplete gamma-function [[#References|[1]]], [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Pagurova, "Tables of the incomplete gamma-function" , Moscow (1963) (In Russian) {{MR|0159040}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> K. Pearson (ed.) , ''Tables of the incomplete gamma function'' , Cambridge Univ. Press (1957)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.L. Johnson, S. Kotz, "Distributions in statistics" , '''1. Continuous univariate distributions''' , Wiley (1970) {{MR|0270476}} {{MR|0270475}} {{ZBL|0213.21101}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> L.J. Comrie, "Chambers's six-figure mathematical tables" , '''II''' , Chambers (1949)</TD></TR></table>