Beta distribution

The Beta distribution is defined by a density on the interval [0,1] whose shape is determined by two positive parameters, α and β. For 0 < x < 1, its density is f(x) = x^α−1(1−x)^β−1/B(α,β), where B is the beta function that normalizes the curve.¹

The parameters are shape parameters rather than direct lower and upper bounds. When α = β, the distribution is symmetric; α = β = 1 gives the uniform distribution, while α and β greater than 1 produce a mound-shaped curve. Values below 1 create singular behavior near an endpoint: α < 1 emphasizes zero and β < 1 emphasizes one. The support is bounded, so the distribution is suited to quantities that must remain proportions or probabilities.

The mean of a Beta distribution is α/(α+β), and its variance is αβ/[(α+β)²(α+β+1)].¹ These formulas separate location from concentration: keeping the mean fixed while increasing α+β makes the distribution more tightly concentrated around that mean.

Several familiar shapes are special cases. Beta(1,1) is uniform; Beta(1,β) and Beta(α,1) are one-sided power-law forms; and Beta(1/2,1/2), the arcsine distribution, places substantial mass near both endpoints. The distribution is also closed under reflection: if X has Beta(α,β) distribution, then 1−X has Beta(β,α) distribution. Its cumulative distribution function is expressed through the regularized incomplete beta function, which is commonly evaluated numerically by statistical software.²

The Beta distribution is the conjugate prior for the success probability of a binomial experiment. With a Beta(α,β) prior and observed successes and failures, the posterior remains Beta, with the two parameters increased by the corresponding counts.

This relationship supports probability estimation in areas such as reliability, medical testing, polling, and A/B experiments, although the prior parameters encode assumptions and should not be treated as automatically objective. In frequentist work, the distribution can model a population proportion or serve as a flexible bounded approximation. Parameter estimation may use moments, maximum likelihood, or numerical methods; likelihood equations generally require solving digamma-function relationships rather than applying a simple closed-form formula.³

The Beta distribution is a special case of the Dirichlet distribution, obtained when there are only two categories; this makes it a natural model for a binary composition.⁴ It also appears in order statistics, where suitably standardized sample order values have beta laws, and in random-generation methods based on gamma variables.

Endpoint behavior deserves care in computation. If either parameter is below one, the density can become unbounded at an endpoint even though the total probability remains finite; a density height is not itself a probability. Software libraries therefore calculate probabilities through specialized incomplete-beta algorithms rather than relying on direct plotting or naive integration.² For data that include exact zeros or ones, a plain continuous Beta model may be inadequate, and a zero-one-inflated or hurdle construction can be more appropriate.

Beta function

The normalizing function B(α,β) that makes the Beta density integrate to one.

Conjugate prior

A prior distribution whose posterior has the same mathematical family after data are observed.

Shape parameter

A parameter controlling the form of a distribution, including its concentration and skewness.

Arcsine distribution

The Beta(1/2,1/2) distribution, with density that is largest near both endpoints.

Dirichlet distribution

A multivariate generalization of the Beta distribution for proportions across several categories.