Beta Distribution
X ~ Beta(α, β)
Bounded / Conjugate-Prior Family • Published July 19, 2026
The distribution for anything bounded to (0,1) -- a probability, proportion, or share: history from Euler's Beta function through Thomas Bayes's 1763 essay, the range of shapes (U-shaped, flat, skewed) the two parameters cover, exact ties to two already-published sheets (Beta(1,1) = Uniform, and the k-th order statistic of n iid Uniforms is exactly Beta(k, n-k+1)), its role as the Binomial's conjugate prior, closed-form method-of-moments plus numerical MLE, and simulation via a ratio of two independent Gammas.
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About This Reference Sheet
The distribution for anything bounded to (0,1) -- a probability, proportion, or share: history from Euler's Beta function through Thomas Bayes's 1763 essay, the range of shapes (U-shaped, flat, skewed) the two parameters cover, exact ties to two already-published sheets (Beta(1,1) = Uniform, and the k-th order statistic of n iid Uniforms is exactly Beta(k, n-k+1)), its role as the Binomial's conjugate prior, closed-form method-of-moments plus numerical MLE, and simulation via a ratio of two independent Gammas.
Support
x ∈ (0, 1)
Parameters
α > 0 (shape), β > 0 (shape)