Conway–Maxwell–Poisson distribution

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.

Conway–Maxwell–Poisson

Probability mass function
Cumulative distribution function
Parameters	${\displaystyle \lambda >0,\nu \geq 0}$
Support	${\displaystyle x\in \{0,1,2,\dots \}}$
PMF	${\displaystyle {\frac {\lambda ^{x}}{(x!)^{\nu }}}{\frac {1}{Z(\lambda ,\nu )}}}$
CDF	${\displaystyle \sum _{i=0}^{x}\Pr(X=i)}$
Mean	${\displaystyle \sum _{j=0}^{\infty }{\frac {j\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}}$
Median	No closed form
Mode	See text
Variance	${\displaystyle \sum _{j=0}^{\infty }{\frac {j^{2}\lambda ^{j}}{(j!)^{\nu }Z(\lambda ,\nu )}}-\operatorname {mean} ^{2}}$
Skewness	Not listed
Ex. kurtosis	Not listed
Entropy	Not listed
MGF	${\displaystyle {\frac {Z(e^{t}\lambda ,\nu )}{Z(\lambda ,\nu )}}}$
CF	${\displaystyle {\frac {Z(e^{it}\lambda ,\nu )}{Z(\lambda ,\nu )}}}$
PGF	${\displaystyle {\frac {Z(t\lambda ,\nu )}{Z(\lambda ,\nu )}}}$