ConwayMaxwellPoissonVerteilung

The Conway-Maxwell-Poisson distribution (CMP) is a two-parameter discrete probability distribution for nonnegative integers that generalizes the Poisson distribution to accommodate under- or over-dispersion in count data. It is used when the Poisson assumption of equal mean and variance is too restrictive.

The probability mass function is P(X = k) = (lambda^k) / ((k!)^nu Z(lambda, nu)) for k = 0, 1, 2, ..., where lambda > 0 and nu > 0. The normalization constant Z(lambda, nu) = sum_{j=0}^∞ (lambda^j) / (j!)^nu ensures that the probabilities sum to one. When nu = 1, the CMP reduces to the standard Poisson distribution with mean and variance both equal to lambda. For nu > 1 the distribution is more concentrated (under-dispersed) than Poisson, while for nu < 1 it is more dispersed (over-dispersed).

Mean and variance do not have closed-form expressions; they are derived from the normalization constant. In particular, E[X] = lambda * ∂ log Z(lambda, nu) / ∂ lambda, and Var[X] = lambda^2 * ∂^2 log Z(lambda, nu) / ∂ lambda^2. Z(lambda, nu) must be computed numerically, typically by truncating the infinite sum and using numerical methods for the derivatives. Parameter estimation is commonly performed via maximum likelihood or Bayesian approaches, which require repeated evaluation of Z and its derivatives.

Applications of the CMP include modeling count data in fields such as biology, ecology, epidemiology, and queueing theory, where dispersion differs from Poisson. The CMP framework also supports regression extensions, enabling covariate effects to influence either the rate parameter or the dispersion parameter, providing a flexible tool for complex count data.