Beta-Binomial Distribution
X ~ BetaBinom(n, α, β)
Beta-Binomial Mixture Family • Published July 20, 2026
The overdispersion fix for Binomial count data: history from Yule and Pearson's non-independent-trials work, the exact Beta-Binomial mixture representation that parallels the already-published Negative Binomial's Gamma-Poisson mixture, exact ties to two already-published sheets (α=β=1 recovers the Discrete Uniform; α,β→∞ recovers the ordinary Binomial), a Fisher information matrix expressed as a finite sum with no simpler closed form, a coupled two-parameter digamma maximum-likelihood system verified against direct optimization, and simulation via the Beta-mixing construction.
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About This Reference Sheet
The overdispersion fix for Binomial count data: history from Yule and Pearson's non-independent-trials work, the exact Beta-Binomial mixture representation that parallels the already-published Negative Binomial's Gamma-Poisson mixture, exact ties to two already-published sheets (α=β=1 recovers the Discrete Uniform; α,β→∞ recovers the ordinary Binomial), a Fisher information matrix expressed as a finite sum with no simpler closed form, a coupled two-parameter digamma maximum-likelihood system verified against direct optimization, and simulation via the Beta-mixing construction.
Support
x ∈ {0, 1, ..., n}
Parameters
n ≥ 1 integer, α > 0, β > 0