Given a Poisson distribution with a rate of change 
,
the distribution function 
giving the waiting times until the 
th Poisson event is
 |  |  |
(1)
|
 |  |  |
(2)
|
for 
,
where 
is a complete gamma function, and 
an incomplete
gamma function. With 
explicitly an integer, this distribution is known as the Erlang
distribution, and has probability function
 |
(3)
|
It is closely related to the gamma distribution, which is obtained by letting 
(not necessarily an integer) and defining 
. When 
, it simplifies to the exponential
distribution.
Evans et al. (2000, p. 71) write the distribution using the variables 
and 
.
