Displaced Poisson distribution

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.

Probability mass function
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 $$I\left(r,\lambda\right)$$ is the Pearson's incomplete gamma function:

I(r,\lambda)=\sum^\infty_{y=r}\frac{e^{-\lambda} \lambda^y}{y!}, $$ where s is the integral part of r. The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is $$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)$$.

Examples
One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal. The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:
 * the distribution of insect populations in crop fields;
 * the number of flowers on plants;
 * motor vehicle crash counts; and
 * word or sentence lengths in writing.

Descriptive Statistics

 * For a displaced Poisson-distributed random variable, the mean is equal to $$\lambda - r$$ and the variance is equal to $$\lambda$$.
 * The mode of a displaced Poisson-distributed random variable are the integer values bounded by $$\lambda - r - 1$$ and $$\lambda - r$$ when $$\lambda \geq r+1$$. When $$\lambda < r+1$$, there is a single mode at $$x=0$$.
 * The first cumulant $$\kappa_{1}$$ is equal to $$\lambda - r$$ and all subsequent cumulants $$\kappa_{n}, n \geq 2$$ are equal to $$\lambda$$.