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): 643654. doi:10.1080/01621459.1967.10482938.



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