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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 λ > 0 {\displaystyle \lambda >0} and r is a new parameter; the Poisson distribution is recovered at r = 0. Here I {\displaystyle I\left} is the Pearson's incomplete gamma function:
where s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution / P {\displaystyle P/P} ] is given by λ / n {\displaystyle \lambda /n} for n > 0 {\displaystyle n>0} and the displaced Poisson generalizes this ratio to λ / {\displaystyle \lambda /\left}.