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 .