Beta Distribution Calculator

The beta distribution is one of the most mathematically rich distributions in probability theory — its PDF involves the beta function (a generalization of the factorial through the gamma function), its CDF requires the regularized incomplete beta function, and its moments have elegant closed forms. The beta distribution calculator for mathematics courses provides the full analytical framework alongside numerical results.

The PDF: Mathematical Definition

The probability density function of Beta(α, β) for x in [0, 1] with α, β > 0:

f(x; α, β) = x^(α−1) × (1−x)^(β−1) / B(α, β)

where the beta function provides the normalization:

B(α, β) = Γ(α) × Γ(β) / Γ(α + β)

and Γ(n) = (n−1)! for positive integers, or the complete gamma function for non-integer values. The logarithmic form used in computation: ln f(x) = (α−1)ln(x) + (β−1)ln(1−x) − ln B(α,β). This log-form avoids numerical underflow for extreme parameter values. Use this online calculator for any α, β, and x. The applied beta distribution calculator covers Bayesian and A/B testing applications.

Moment Generating Function and Moments

The raw moments of the beta distribution have elegant closed forms:

• Mean (1st moment): E[X] = α / (α + β)
• Second moment: E[X²] = α(α+1) / [(α+β)(α+β+1)]
• Variance: Var(X) = E[X²] − (E[X])² = αβ / [(α+β)²(α+β+1)]
• Skewness: 2(β−α)√(α+β+1) / [(α+β+2)√(αβ)]
• Excess kurtosis: 6[(α−β)²(α+β+1) − αβ(α+β+2)] / [αβ(α+β+2)(α+β+3)]

The moment generating function does not exist in closed form for the beta distribution; the characteristic function is expressible as a confluent hypergeometric function.

The Regularized Incomplete Beta Function: CDF Computation

The CDF requires the regularized incomplete beta function I_x(α, β):

F(x; α, β) = I_x(α, β) = B(x; α, β) / B(α, β)

where B(x; α, β) = integral from 0 to x of t^(α−1)(1−t)^(β−1) dt is the incomplete beta function. This cannot be expressed in closed form for general α, β — numerical methods (continued fraction expansion, quadrature) are required for evaluation. The regularized incomplete beta function is available in all major mathematical software: Python's scipy.special.betainc, R's pbeta(), MATLAB's betainc(). The relationship to the F-distribution CDF and the binomial CDF makes the incomplete beta function one of the most important special functions in statistics. The binomial distribution calculator and probability distribution calculators cover related statistical distributions.

Relationship to Other Distributions

The beta distribution is the parent of several important distributions and limits to others:

• Beta(1, 1) = Uniform(0, 1): the flat prior; both shape parameters at their minimum integer value
• Beta(1/2, 1/2) = Arcsine distribution: models the fraction of time a random walk spends positive
• Beta(n, m) → Normal as n, m → ∞: central limit theorem convergence
• F-distribution connection: if X ~ Beta(α, β), then X/(1−X) × (β/α) ~ F(2α, 2β)
• Beta-binomial distribution: mixing a binomial likelihood with a beta prior produces a beta-binomial marginal distribution — used for overdispersed count data