betabinomial

The beta-binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials when the probability of success in each trial is itself random and follows a beta distribution. It arises as a compound distribution: if the success probability p is drawn from a Beta(alpha, beta) distribution and then K is drawn from a Binomial(n, p) distribution, the marginal distribution of K is beta-binomial.

Its probability mass function is P(K = k) = binomial(n, k) times Beta(k + alpha, n - k + beta) divided by Beta(alpha, beta), for k = 0, 1, ..., n. Here n is the number of trials, alpha and beta are positive shape parameters of the beta prior, and Beta denotes the beta function. The beta-binomial thus generalizes the binomial distribution by allowing extra variation or overdispersion in the success probability.

Key moments include the mean and variance. The mean is E[K] = n * alpha / (alpha + beta). The variance can be written as Var(K) = n * p * (1 - p) * (n + alpha + beta) / (alpha + beta + 1), where p = alpha / (alpha + beta). An equivalent form is Var(K) = n * alpha * beta * (n + alpha + beta) / [(alpha + beta)^2 * (alpha + beta + 1)]. As alpha and beta grow large with a fixed mean p, the beta-binomial converges to a binomial distribution with parameter p.

Applications include modeling overdispersion in binomial data, Bayesian hierarchical modeling, and fields such as ecology, genetics, quality control, and marketing. The beta-binomial is related to the Dirichlet-multinomial in higher dimensions, serving as the two-category case.