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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
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<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>
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{{TEX|auto}}
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{{TEX|done}}
  
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:
+
A continuous probability distribution concentrated on the positive semi-axis  $  0 < x < \infty $
 +
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/g0433005.png" /></td> </tr></table>
+
$$
 +
g _  \alpha  ( x)  = \
 +
{
 +
\frac{1}{\Gamma ( \alpha ) }
 +
}
 +
x ^ {\alpha - 1 } e ^ {- x } ,
 +
$$
  
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
+
where  $  \alpha $
 +
is a parameter assuming positive values, and $  \Gamma ( \alpha ) $
 +
is Euler's gamma-function:
  
<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/g0433008.png" /></td> </tr></table>
+
$$
 +
\Gamma ( \alpha )  = \
 +
\int\limits _ { 0 } ^  \infty 
 +
y ^ {\alpha - 1 }
 +
e ^ {- y }  dy.
 +
$$
  
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
+
The corresponding distribution function for  $  x \leq  0 $
 +
is zero, and for $  x > 0 $
 +
it is expressed by the formula
  
<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/g04330018.png" /></td> </tr></table>
+
$$
 +
G _  \alpha  ( x)  = \
 +
{
 +
\frac{1}{\Gamma ( \alpha ) }
 +
}
 +
\int\limits _ { 0 } ^ { x }
 +
y ^ {\alpha - 1 }
 +
e ^ {- y }  dy.
 +
$$
 +
 
 +
The integral on the right-hand side is called the incomplete gamma-function. The density  $  g _  \alpha  ( x) $
 +
is unimodal and for  $  \alpha > 1 $
 +
it attains the maximum  $  ( \alpha - 1) ^ {\alpha - 1 } e ^ {- ( \alpha - 1 ) } / \Gamma ( \alpha ) $
 +
at the point  $  x = \alpha - 1 $.  
 +
If  $  0 < \alpha < 1 $
 +
the density  $  g _  \alpha  ( x) $
 +
decreases monotonically with increasing  $  x $,
 +
and if  $  x \downarrow 0 $,
 +
$  g _  \alpha  ( x) $
 +
increases without limit. The characteristic function of the gamma-distribution has the form
 +
 
 +
$$
 +
\phi ( t)  =  ( 1 - it) ^ {- \alpha } .
 +
$$
  
 
The moments of the gamma-distribution are given by the formula
 
The moments of the gamma-distribution are given by the formula
  
<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/g04330019.png" /></td> </tr></table>
+
$$
 +
m _ {k}  = \int\limits _ { 0 } ^  \infty 
 +
x  ^ {k} g _  \alpha  ( x)  dx  = \
 +
 
 +
\frac{\Gamma ( \alpha + k) }{\Gamma ( \alpha ) }
 +
,\ \
 +
k > - \alpha .
 +
$$
 +
 
 +
In particular, the mathematical expectation and variance are equal to  $  \alpha $.
 +
The set of gamma-distributions is closed with respect to the operation of convolution:
 +
 
 +
$$
 +
g _ {\alpha _ {1}  } \star g _ {\alpha _ {2}  }  = \
 +
g _ {\alpha _ {1}  + \alpha _ {2} } .
 +
$$
  
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:
+
Gamma-distributions play a significant, though not always an explicit, role in applications. In the particular case of  $  \alpha = 1 $
 +
one obtains the exponential density. In queueing theory, the gamma-distribution for an  $  \alpha $
 +
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  $  \chi  ^ {2} = X _ {1}  ^ {2} + \dots + X _ {n}  ^ {2} $
 +
of independent  $  ( 0, 1) $
 +
normally-distributed random variables has density  $  g _ {n/2} ( x/2) /2 $
 +
and is known as the "chi-squared" distribution with  $  n $
 +
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  $  F $-
 +
distribution]] and the [[Fisher z-distribution|Fisher  $  z $-
 +
distribution]]). If  $  X _ {1} $
 +
and  $  X _ {2} $
 +
are independent and are distributed with densities  $  g _ {\alpha _ {1}  } $
 +
and  $  g _ {\alpha _ {2}  } $,
 +
then the random variable  $  X _ {1} / ( X _ {1} + X _ {2} ) $
 +
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/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-z-distribution(2)|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
+
\frac{\Gamma ( \alpha _ {1} + \alpha _ {2} ) }{\Gamma ( \alpha _ {1} ) \Gamma ( \alpha _ {2} ) }
  
<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>
+
x ^ {\alpha _ {1} - 1 }
 +
( 1 - x) ^ {\alpha _ {2} - 1 } ,\ \
 +
0 < x < 1,
 +
$$
  
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]]].
+
which is known as the density of the beta-distribution. The densities of linear functions $  aX + b $
 +
of random variables $  X $
 +
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:41, 5 June 2020


A continuous probability distribution concentrated on the positive semi-axis $ 0 < x < \infty $ with density

$$ g _ \alpha ( x) = \ { \frac{1}{\Gamma ( \alpha ) } } x ^ {\alpha - 1 } e ^ {- x } , $$

where $ \alpha $ is a parameter assuming positive values, and $ \Gamma ( \alpha ) $ is Euler's gamma-function:

$$ \Gamma ( \alpha ) = \ \int\limits _ { 0 } ^ \infty y ^ {\alpha - 1 } e ^ {- y } dy. $$

The corresponding distribution function for $ x \leq 0 $ is zero, and for $ x > 0 $ it is expressed by the formula

$$ G _ \alpha ( x) = \ { \frac{1}{\Gamma ( \alpha ) } } \int\limits _ { 0 } ^ { x } y ^ {\alpha - 1 } e ^ {- y } dy. $$

The integral on the right-hand side is called the incomplete gamma-function. The density $ g _ \alpha ( x) $ is unimodal and for $ \alpha > 1 $ it attains the maximum $ ( \alpha - 1) ^ {\alpha - 1 } e ^ {- ( \alpha - 1 ) } / \Gamma ( \alpha ) $ at the point $ x = \alpha - 1 $. If $ 0 < \alpha < 1 $ the density $ g _ \alpha ( x) $ decreases monotonically with increasing $ x $, and if $ x \downarrow 0 $, $ g _ \alpha ( x) $ increases without limit. The characteristic function of the gamma-distribution has the form

$$ \phi ( t) = ( 1 - it) ^ {- \alpha } . $$

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

$$ m _ {k} = \int\limits _ { 0 } ^ \infty x ^ {k} g _ \alpha ( x) dx = \ \frac{\Gamma ( \alpha + k) }{\Gamma ( \alpha ) } ,\ \ k > - \alpha . $$

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

$$ g _ {\alpha _ {1} } \star g _ {\alpha _ {2} } = \ g _ {\alpha _ {1} + \alpha _ {2} } . $$

Gamma-distributions play a significant, though not always an explicit, role in applications. In the particular case of $ \alpha = 1 $ one obtains the exponential density. In queueing theory, the gamma-distribution for an $ \alpha $ 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 $ \chi ^ {2} = X _ {1} ^ {2} + \dots + X _ {n} ^ {2} $ of independent $ ( 0, 1) $ normally-distributed random variables has density $ g _ {n/2} ( x/2) /2 $ and is known as the "chi-squared" distribution with $ n $ 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 $ F $- distribution and the Fisher $ z $- distribution). If $ X _ {1} $ and $ X _ {2} $ are independent and are distributed with densities $ g _ {\alpha _ {1} } $ and $ g _ {\alpha _ {2} } $, then the random variable $ X _ {1} / ( X _ {1} + X _ {2} ) $ has density

$$ \frac{\Gamma ( \alpha _ {1} + \alpha _ {2} ) }{\Gamma ( \alpha _ {1} ) \Gamma ( \alpha _ {2} ) } x ^ {\alpha _ {1} - 1 } ( 1 - x) ^ {\alpha _ {2} - 1 } ,\ \ 0 < x < 1, $$

which is known as the density of the beta-distribution. The densities of linear functions $ aX + b $ of random variables $ X $ 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://encyclopediaofmath.org/index.php?title=Gamma-distribution&oldid=24074
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article