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The gamma distribution

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The gamma distribution is a two-parameter family of continuous probability distributions. Two different parameterisations are in common use, see below, with the  parameterisation being apparently somewhat more common in econometrics and the  parameterisation being somewhat more common in Bayesian statistics.

 

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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.

 

Nematrian web functions

 

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “gamma”. Functions relating to a generalised version of this distribution including an additional location (i.e. shift) parameter may be accessed by using a DistributionName of “gamma3”, see also including additional shift and scale parameters. For details of other supported probability distributions see here.

 


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