Beta Distribution Calculator

The beta distribution is a continuous probability distribution defined on the interval from 0 to 1, which makes it the natural model for a proportion, a rate or a probability. Its shape is set by two positive parameters, alpha and beta, often written as the number of prior successes and failures. When alpha and beta are equal the distribution is symmetric around 0.5, when alpha is larger the weight shifts towards 1, and when beta is larger the weight shifts towards 0. This calculator takes your alpha and beta and returns the summary statistics of the distribution: the mean, the variance and standard deviation, the mode where the density peaks, and the skewness that measures how lopsided the curve is. It also evaluates the probability density function (PDF) and the cumulative distribution function (CDF) at any point x you choose between 0 and 1, so you can read off the height of the curve and the probability that a random draw falls at or below that point. The beta distribution turns up constantly in Bayesian statistics as the conjugate prior for the binomial, in project scheduling under the PERT method, and in modelling conversion rates, click-through rates and component reliability. Enter alpha, beta and a value of x, and the results update as you type. All inputs must be positive, and x must sit between 0 and 1.

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At x = 0.30 the probability density f(x) is 2.1609 and the cumulative probability P(X ≤ x) is 0.5798. Distribution skewness is 0.5963.

Alpha and beta must be positive and x must be between 0 and 1. The mode is defined at an interior peak only when both alpha and beta are greater than 1.