Gamma-distribution

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)
[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)