Table 1 Well-known distributions associated with the Conway-Maxwell-Poisson (CMP) distribution for special cases of λ and ν

ν=1

e ^λ

\(P(X=x) = \frac {e^{-\lambda }\lambda ^{x}}{x!}, \ x=0,1,2,\ldots \)

Poisson(λ)

ν=0,λ<1

\(\frac {1}{1-\lambda }\)

P(X=x)=(1−λ)λ ^x, x=0,1,2,…

Geom(1−λ)

ν→∞

1+λ

\(P(X=0) = \frac {1}{1+\lambda }; \ P(X=1) = \frac {\lambda }{1+\lambda }\)

Bernoulli\(\left (\frac {\lambda }{1+\lambda }\right)\)