The gamma distribution
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The gamma distribution is a twoparameter 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.
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 chisquared
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 MaxwellBoltzmann
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 noncentral 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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