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

From: A flexible distribution class for count data

Case Z(λ,ν) pmf Distribution
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\(\left (\frac {\lambda }{1+\lambda }\right)\)