Displaced Poisson distribution
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 formula_2 and "r" is a new parameter; the Poisson distribution is recovered at "r" = 0. Here formula_3 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 (that is formula_5) is given by formula_6 for formula_7 and the displaced Poisson generalizes this ratio to formula_8.