Given a Poisson distribution with a rate of change , the distribution function giving the waiting times until the th Poisson event is
(1)
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(2)
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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)
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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 .