Distribution name

Gamma distribution

Common notation

Parameters

Has two commonly used parameterisations:

 = shape parameter ()

 = scale parameter () or  = inverse scale (i.e. rate) parameter () where .

Unless otherwise specified the material below assumes the first parameterisation (i.e. using a scale parameter)

Domain

Probability density function

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

Moment generating function

Characteristic function

Other comments

The gamma distribution can also be defined with a location parameter, , say, in which case its domain is shifted to  .

Its mode is  for .

If  follows an exponential distribution with rate parameter  then  .

If  follows a chi-squared distribution, with  degrees of freedom, i.e. i.e.   then  and .

If  is integral then  is also called the Erlang distribution. It is the distribution of the sum of  independent exponential variables each with mean . Events that occur independently with some average rate are commonly modelled using a Poisson process. The waiting times between  occurrences of the event are then Erlang distributed whilst the number of events in a given amount of time is Poisson distributed.

If  follows a Maxwell-Boltzmann distribution with parameter  then  . If  follows a skew logistic distribution with parameter  then .

The gamma distribution is the conjugate prior for the precision (i.e. inverse variance) of a normal distribution and for the exponential distribution.

The gamma distribution has the ‘summation’ property that if  for  and the  are independent then .

Its non-central moments ( are . There is in general no simple closed form for its median.