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 
.
