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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|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

$$ 

\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|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>