Difference between revisions of "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)