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

${\displaystyle P(X=n)={\begin{cases}e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r,\lambda \right)}},\quad n=0,1,2,\ldots &{\text{if }}r\geq 0\\[10pt]e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r+s,\lambda \right)}},\quad n=s,s+1,s+2,\ldots &{\text{otherwise}}\end{cases}}}$

where ${\displaystyle \lambda >0}$ and r is a new parameter; the Poisson distribution is recovered at r = 0. Here ${\displaystyle I\left(r,\lambda \right)}$ is the Pearson's incomplete gamma function:

${\displaystyle I(r,\lambda )=\sum _{y=r}^{\infty }{\frac {e^{-\lambda }\lambda ^{y}}{y!}},}$

where 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 ${\displaystyle P(X=n)/P(X=n-1)}$) is given by ${\displaystyle \lambda /n}$ for ${\displaystyle n>0}$ and the displaced Poisson generalizes this ratio to ${\displaystyle \lambda /\left(n+r\right)}$.

References

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.