Displaced Poisson distribution information

Displaced Poisson Distribution
	Probability mass function Displaced Poisson distributions for several values of and . At , the Poisson distribution is recovered. The probability mass function is only defined at integer values.
Parameters	,
Support
Mean
Mode
Variance
MGF	, When is a negative integer, this becomes

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

Probability mass function Displaced Poisson distributions for several values of $\lambda$ and $r$ . At $r=0$ , the Poisson distribution is recovered. The probability mass function is only defined at integer values.
Parameters	$\lambda \in (0,\infty )$ , $r\in (-\infty ,\infty )$
Support	$k\in \mathbb {N} _{0}$
Mean	$\lambda -r$
Mode	${\begin{cases}\left\lceil \lambda -r\right\rceil -1,\left\lfloor \lambda -r\right\rfloor &{\text{if }}\lambda \geq r+1\\0&{\text{if }}\lambda <r+1\\\end{cases}}$
Variance	$\lambda$
MGF	$e^{\lambda \left(e^{t-1}\right)-tr}\cdot {\dfrac {I\left(r+s,\lambda e^{t}\right)}{I\left(r+s,\lambda \right)}}$ , $I\left(r,\lambda \right)=\sum _{y=r}^{\infty }{\dfrac {e^{-\lambda }\lambda ^{y}}{y!}}$ When $r$ is a negative integer, this becomes $e^{\lambda \left(e^{t-1}\right)-tr}$

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