< List of probability distributions < Conway–Maxwell–Poisson distribution
What is the Conway–Maxwell–Poisson distribution?
The Conway–Maxwell–Poisson distribution (also called the CMP or COM–Poisson) is a two-parameter generalization of the Poisson distribution that can account for overdispersion or underdispersion in data. It is a valuable tool in several fields, including economics, biology, and epidemiology.
The CMP distribution, introduced by Richard Conway and William Maxwell , was developed to address limitations with the Poisson distribution — a common probability distribution used to model the number of occurrences of an event in a given time or space. The Poisson distribution has the assumption of homogeneity, which means that the variance of the distribution is equal to the mean. This assumption leads to overdispersion, where the variance is greater than the mean, or underdispersion, where the variance is less than the mean.
The Conway–Maxwell–Poisson distribution adds an additional parameter (ν) to the Poisson, to model the tail behavior and account for overdispersion or underdispersion. A random variable X follows a Conway–Maxwell–Poisson distribution if its probability mass function (PMF) is 
The extra parameter ν is called the dispersion parameter, and it can take any value greater than or equal to zero.
- If ν = 1, the ratio equals the Poisson distribution rate of decay.
- If ν < 1, the rate of decay decreases, which enables modeling overdispersed processes with longer tails than the Poisson distribution.
- If ν > 1, the rate of decay increases and the tail shortens, accommodating underdispersed data.
The probability density function (PDF) is 
P(Y = y; λ, ν = 0) = ( 1 – λ) λy
A closed form of the cumulative distribution function (CDF) of X does not exist. However, if the dispersion parameter ν is an integer, the following formula — in terms of the generalized hypergeometric function — can be used:
Relationship to other distributions
The Conway–Maxwell–Poisson distribution is a member of the exponential family of distributions.
- When ν = 1, X ∼ CMP(λ, 1), the formula becomes the Poisson distribution Po(λ) with normalizing constant Z(λ, 1) = eλ.
- Where ν = 0 and 0 < λ < 1, the formula becomes a geometric distribution, with Z(λ, 0) = (1−λ) −1 .
- In the limit ν → ∞, the random variable X converges in distribution to a Bernoulli random variable, with mean λ(1 + λ) −1 and limν→∞ Z(λ, ν) = 1 + λ . In this case, the data can only take on two values (0, 1), so is suited for modeling extreme underdispersion.
Applications of the Conway–Maxwell–Poisson distribution
The CMP distribution was originally proposed by Conway and Maxwell in 1962  to handle queueing systems with state-dependent service rates. Since then, it has evolved to fit many real-life applications and is used in various fields such as economics, biology, and demography.
One example is in modeling the number of worker absences for a given day. The Poisson distribution may not accurately model the data since some employees may have a higher likelihood of missing work than others. This is where the Conway–Maxwell–Poisson distribution with a dispersion parameter can provide a better fit.
Another example is in disease outbreak modeling. When there is an outbreak, there may be individuals with different susceptibilities or recoveries, leading to heterogeneity in the population. The Conway–Maxwell–Poisson distribution can account for the heterogeneity in the population and provide a more accurate model.
Further reading on applications:
- Distribution theory: Shmueli et al., 2005 .
- Control chart theory: Sellers, 2011 ,
- Multivariate data analysis: Sellers and Balakrishnan, 2016 ,
- Regression analysis: Sellers and Shmueli, 2010  includes COMPoissonReg package in R.
- Risk analysis: Guikema and Coffelt, 2008 .
 R. W. Conway and W. L. Maxwell. A queueing model with state dependent service rate. J. Ind. Engineering 12, 132–136 (1962).
 Daly, F. & Gaunt, R. The Conway-Maxwell-Poisson distribution:
Distributional theory and approximation. ALEA, Lat. Am. J. Probab. Math. Stat. 13, 635–658 (2016) DOI: 10.30757/ALEA.v13-25
 SAS Help Center. PDF Conway-Maxwell-Poisson Distribution Function. Retrieved April 13, 2021 from: https://documentation.sas.com/doc/en/pgmsascdc/v_017/ds2ref/n1bwtppmvqmdlgn1rx99e8qchvof.htm
 Nadarajah, S. “Useful moment and CDF formulations for the COM–Poisson distribution.” Statistical Papers 50 (2009): 617–622.
 Shmueli G, Minka TP, Kadane JB, Borle S, Boatwright P (2005). A useful distribution for fitting discrete data: revival of the Conway-Maxwell-Poisson distribution. Applied Statistics, 54:127-142.
 Sellers KF (2011) A generalized statistical control chart for over- or under-dispersed data, Quality Reliability Engineering International, 28 (1), 59-65.
 Sellers, KF, Morris, DS, Balakrishnan, N: Bivariate Conway-Maxwell-Poisson distribution: Formulation, properties, and inference. J. Multivar. Anal. 150, 152–168 (2016)
 Kimberly F. Sellers & Galit Shmueli (2010). A Flexible Regression Model for Count Data. Annals of Applied Statistics, 4(2), 943-961.
 Guikema, S.D., Coffelt, J.P. (2008). Modeling Count Data in Risk Analysis and Reliability Engineering. In: Misra, K.B. (eds) Handbook of Performability Engineering. Springer, London. https://doi.org/10.1007/978-1-84800-131-2_37