Conway–Maxwell–Poisson distribution

Conway–Maxwell–Poisson
Parameters	$\lambda >0,\nu \geq 0$ $\lambda >0,\nu \geq 0$
Support	$x\in \{0,1,2,\dots \}$ $x\in \{0,1,2,\dots \}$
pmf	${\frac {\lambda ^{x}}{(x!)^{\nu }}}{\frac {1}{Z(\lambda ,\nu )}}$ ${\frac {\lambda ^{x}}{(x!)^{\nu }}}{\frac {1}{Z(\lambda ,\nu )}}$
CDF	$\sum _{i=0}^{x}\mathbb {P} (X=i)$ $\sum _{{i=0}}^{x}{\mathbb {P}}(X=i)$
Mean	$\sum _{j=0}^{\infty }{\frac {j\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}$ $\sum _{{j=0}}^{\infty }{\frac {j\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}$
Median	No closed form
Mode	Not listed
Variance	$\sum _{j=0}^{\infty }{\frac {j^{2}\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}-{\text{Mean}}^{2}$ $\sum _{{j=0}}^{\infty }{\frac {j^{2}\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}-{\text{Mean}}^{2}$
Skewness	Not listed
Ex. kurtosis	Not listed
Entropy	Not listed
MGF	Not listed
CF	Not listed

In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM-Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.

The COM-Poisson distribution was originally proposed by Conway and Maxwell in 1962 as a solution to handling queueing systems with state-dependent service rates. The probabilistic and statistical properties of the distribution were published by Shmueli et al. (2005).

The COM-Poisson is defined to be the distribution with probability mass function

for x = 0,1,2,..., $\lambda >0$ $\lambda >0$ and $\nu$ $\nu$ ≥ 0, where

...
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