ConwayMaxwellbinomial

The Conway-Maxwell-binomial distribution, commonly abbreviated as the CMB distribution, is a three-parameter discrete distribution on the integers k = 0, 1, ..., n. It generalizes the binomial distribution by introducing a dispersion parameter nu in addition to the number of trials n and the success probability p. For fixed n, p in (0,1), and nu > 0, its probability mass function is P(X = k) = [binom(n, k) p^k (1-p)^(n-k) nu^(k(n-k))] / Z, where Z = sum_{j=0}^n binom(n, j) p^j (1-p)^(n-j) nu^(j(n-j)) is a normalizing constant. When nu = 1, the nu^(k(n-k)) factor equals 1 for all k, and the distribution collapses to the standard binomial(n, p). Thus the CMB provides a flexible extension of the binomial distribution.

The parameter nu controls dispersion: nu > 1 biases probability mass toward midrange values of k near n/2, producing over-dispersion relative to the binomial, while 0 < nu < 1 biases toward the extremes (near 0 or n), yielding under-dispersion. This makes the CMB suitable for modeling binomial-type data with varying levels of correlation among trials.

Moments of the CMB do not have simple closed-form expressions; they are typically obtained numerically from the normalizing constant Z or via derivatives of its cumulant generating function. Estimation of (n, p, nu) usually relies on maximum likelihood, which requires evaluating Z and its derivatives, making inference more computationally intensive than for the binomial.

Applications include modeling count data with over- or under-dispersion and dependence among trials, such as quality control, genetics, epidemiology, and social science research. The CMB is part of the Conway–Maxwell family of distributions.