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

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

where $\lambda>0$ and r is a new parameter; the Poisson distribution is recovered at r = 0. Here $I\left(\cdot,\cdot\right)$ 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 $P(X=n)/P(X=n-1)$) is given by $\lambda/n$ for $n>0$ and the displaced Poisson generalizes this ratio to $\lambda/\left(n+r\right)$.

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.