Conway-Maxwell-Poisson Distribution
X ~ CMP(λ, ν)
Overdispersed Count Family • Published July 22, 2026
The one distribution in this library that moves in either direction from the Poisson's Var = E baseline through a single dispersion parameter: history from Conway and Maxwell's 1962 state-dependent queueing model to its 2005 popularization for count regression, an intractable normalizing constant with no closed form except at the Poisson-recovering ν=1 case, large-λ asymptotic mean/variance approximations checked against direct summation, and simulation via truncated inverse-CDF sampling.
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About This Reference Sheet
The one distribution in this library that moves in either direction from the Poisson's Var = E baseline through a single dispersion parameter: history from Conway and Maxwell's 1962 state-dependent queueing model to its 2005 popularization for count regression, an intractable normalizing constant with no closed form except at the Poisson-recovering ν=1 case, large-λ asymptotic mean/variance approximations checked against direct summation, and simulation via truncated inverse-CDF sampling.
Support
x ∈ {0, 1, 2, ...}
Parameters
λ > 0 (rate-like parameter), ν ≥ 0 (dispersion parameter)