gamma distribution

   kurtosis   = $\frac{6}{k}$ |   entropy    =|   mgf        = $(1 - \theta\,t)^{-k}$  for  $t < 1/\theta$ |   char       = $(1 - \theta\,i\,t)^{-k}$ |

}} In probability theory and statistics, the gamma distribution is a continuous probability distribution.

Specification of the gamma distribution

Probability density function

The probability density function of the gamma distribution can be expressed in terms of the gamma function:

f(x) = x^{k-1} \frac{e^{-x/\theta}}{\Gamma(k)\,\theta^k}

  \ \mathrm{for}\ x > 0 \,\!

where

k > 0

is the shape parameter and

\theta > 0

is the scale parameter of the gamma distribution. (NOTE: this parameterization is what is used in the infobox and the plots.) Alternatively, the gamma distribution can be parameterized in terms of a shape parameter

\alpha = k

and an inverse scale parameter

\beta = 1/\theta

g(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \; x^{\alpha-1} \; \exp(-\beta\,x) \,\!

Cumulative distribution function

The cumulative distribution function can be expressed in terms of the incomplete gamma function,

F(x) = \int_0^x f(u)\,du

   = \frac{\gamma(k, x/\theta)}{\Gamma(k)} \,\!

Properties

X_1

has a gamma distribution with parameters

k_1

and

\theta

, and

X_2

has a gamma distribution with parameters

k_2

and

\theta

, then

X_1 + X_2

has a gamma distribution with parameters

k_1 + k_2

and

\theta

. Or alternatively:

X_1

\mathrm{Gamma}(k_1, \theta)

and

X_2

\mathrm{Gamma}(k_2, \theta)

then

X_1 + X_2

\mathrm{Gamma}(k_1 + k_2, \theta)

The gamma distributions are infinitely divisible probability distributions.

Related distributions

$X$ ~ $\mathrm{Exponential}(\theta)$ is an exponential distribution if $X$ ~ $\mathrm{Gamma}(1, \theta)$ .
$Y$ ~ $\mathrm{Gamma}(N, \theta)$ is a gamma distribution if $Y = X_1 + \cdots + X_N$ and if the $X_i$ ~ $\mathrm{Exponential}(\theta)$ are all independent and share the same parameter $\theta$ .
$X$ ~ $\chi^2(\nu)$ is a chi-square distribution if $X$ ~ $\mathrm{Gamma}(\nu/2, 1/2)$ .
If $k$ is an integer, the gamma distribution is an Erlang distribution (so named in honor of A. K. Erlang) and is the probability distribution of the waiting time of the $k^{\mathrm{th}}$ "arrival" in a one-dimensional Poisson process with intensity $1 / \theta$ .
$X$ ~ $\mathrm{Gamma}(k, \theta)$ then $Y$ ~ $\mathrm{InvGamma}(k, \theta)$ if $Y = 1/X$ , where $\mathrm{InvGamma}$ is the family of inverse-gamma distributions.

References

R. V. Hogg and A. T. Craig. Introduction to Mathematical Statistics, 4th edition. New York: Macmillan, 1978. (See Section 3.3.)