감마 분포는 일반적인 유형의 통계적 분포와 관련된 베타 유통 및 자연적으로 발생한 프로세스에서는 대기 시간 사이의 푸아송되 이벤트가 관련이 있습니다. 감마 분포가 매개변수 표시된
및
,몇 가지의 그림이다.,c0a5aa4824″>
for 
, where 
is a complete gamma function, and 
an incomplete gamma function., 
정수로,이 분포는 Erlang 분포로 알려진 특별한 경우입니다.,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.,
Gammadistribution 함수로 Wolfram 언어로 구현됩니다.,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 
., 둘 다 다음과 같이 도출 될 수있다.,
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 
.,