Beta Distribution Calculator

The beta distribution is defined on the interval [0, 1] and is parameterized by two positive shape parameters, alpha and beta. This makes it a natural model for any quantity that represents a proportion, probability, or fraction.

The PDF is: f(x) = x^(alpha-1) * (1-x)^(beta-1) / B(alpha, beta), where B(alpha, beta) = Gamma(alpha)*Gamma(beta)/Gamma(alpha+beta) is the beta function.

Special cases reveal the distribution’s flexibility: alpha = beta = 1: the uniform distribution on [0, 1]. alpha = beta: a symmetric, bell-shaped distribution centered at 0.5. alpha > beta: skewed left (concentrated toward 1). alpha < beta: skewed right (concentrated toward 0). alpha < 1 or beta < 1: U-shaped or J-shaped distributions with infinite density at one or both endpoints.

The mean is alpha / (alpha + beta). The variance is alpha*beta / ((alpha+beta)^2 * (alpha+beta+1)). The mode, when both alpha and beta exceed 1, is (alpha - 1) / (alpha + beta - 2).

In Bayesian statistics, the beta distribution is the conjugate prior for the Bernoulli and binomial distributions. This means if you start with a Beta(alpha, beta) prior belief about a probability p, and you observe x successes in n trials, the posterior is Beta(alpha + x, beta + n - x). The math stays clean throughout.

Practical uses: click-through rates, conversion rates, test pass rates, proportion of defectives in quality control, and mixing proportions in chemical engineering.

The CDF of the beta distribution is the regularized incomplete beta function I(x; alpha, beta). This function also gives CDF values for the binomial, F, and t distributions, making it one of the most important special functions in applied statistics.