Beta Distribution Calculator

The beta distribution is a continuous probability distribution defined on the interval from zero to one. Two positive shape parameters, conventionally named alpha and beta, control its form. This flexibility makes the distribution useful for modeling probabilities, proportions, conversion rates, reliability, and other quantities with natural lower and upper bounds. When alpha and beta are both one, the density is uniform. Equal values greater than one create a symmetric mound, while unequal values create left or right skew. The probability density function is x raised to alpha minus one, multiplied by one minus x raised to beta minus one, then divided by the beta function B(alpha, beta). Density describes relative concentration near a point; it is not itself the probability of observing that exact continuous value. The cumulative distribution function gives the probability that a beta-distributed random variable is less than or equal to x. This calculator evaluates the CDF using the regularized incomplete beta function and reports both PDF and CDF at the requested point. Several useful summaries follow directly from the shape parameters. The mean is alpha divided by alpha plus beta. The variance is alpha times beta divided by the square of their sum times alpha plus beta plus one. When both parameters exceed one, the interior mode is alpha minus one divided by alpha plus beta minus two. If either parameter is at most one, the most likely behavior occurs at a boundary or can involve two boundaries, so the calculator marks the simple interior mode as undefined. Beta distributions play a central role in Bayesian inference because they are conjugate priors for Bernoulli and binomial likelihoods. A prior Beta(alpha, beta) updated with s successes and f failures becomes Beta(alpha + s, beta + f). This produces interpretable, efficient probability models for rates while expressing uncertainty rather than only a point estimate. The distribution also appears in project modeling, order statistics, and random-number generation. Choose parameters based on the application and remember that alpha and beta are shape values, not probabilities. Use the PDF to compare relative density, the CDF for threshold probability, and the moments to summarize location and spread.