Displaced Poisson distribution

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In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution. The probability mass function is

where and r is a new parameter; the Poisson distribution is recovered at r = 0. Here is the incomplete gamma function and s is the integral part of r. The motivation given by Staff[1] is that the ratio of successive probabilities in the Poisson distribution (that is ) is given by for and the displaced Poisson generalizes this ratio to .


  1. ^ Staff, P. J. (1967). "The displaced Poisson distribution". Journal of the American Statistical Association. 62 (318): 643–654. doi:10.1080/01621459.1967.10482938.