ガンマ分布は、ベータ分布に関連する一般的なタイプの統計分布であり、ポアソン分布イベント間の待ち時間が関連 ガンマ分布には、
と
というラベルの二つの自由パラメータがあります。,c0a5aa4824″>
for 
, where 
is a complete gamma function, and 
an incomplete gamma function., 
整数の場合、この分布はErlang分布と呼ばれる特殊なケースです。,02b6″>
Now let 
(not necessarily an integer) and define 
to be the time between changes., Then the above equation can be written
 |
(13)
|
for 
. This is the probability function for the gamma distribution, and the corresponding distribution function is
 |
(14)
|
where 
is a regularized gamma function.,
これはWolfram言語でGammaDistribution関数として実装されています．,id=”c43570dc99″>
giving moments about 0 of
 |
(19)
|
(Papoulis 1984, p., 147).,iv>
The gamma distribution is closely related to other statistical distributions., If 
, 
, …, 
are independent random variates with a gamma distribution having parameters 
, 
, …,/div>
Also, if 
and 
are independent random variates with a gamma distribution having parameters 
and 
, then 
is a beta distribution variate with parameters 
., いずれも以下のように導出できる。,
where 
is the beta function, which is a beta distribution.,
If 
and 
are gamma variates with parameters 
and 
, the 
is a variate with a beta prime distribution with parameters 
and 
.,iv>
The ratio 
therefore has the distribution
 |
(50)
|
which is a beta prime distribution with parameters 
.,
where 
is the Pochhammer symbol.,0822e6ea8″>
so the cumulants are
 |
(63)
|
If 
is a normal variate with mean 
and standard deviation 
, then
 |
(64)
|
is a standard gamma variate with parameter 
.,