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 .

REFERENCES:

Evans, M.; Hastings, N.; and Peacock, B. "Erlang Distribution." Ch. 12 in Statistical Distributions, 3rd ed. New York: Wiley, pp. 71-73, 2000.