Beta Distribution Calculator
The beta distribution lives entirely on the interval from 0 to 1, which makes it the standard model for probabilities, proportions, and rates. Enter the two shape parameters alpha and beta and an evaluation point x. The calculator returns the probability density at x, the cumulative probability via the regularised incomplete beta function, and the distribution's mean and variance. It is the conjugate prior for the binomial success probability, central to Bayesian A/B testing, and a flexible model for bounded data. The cumulative value uses a continued-fraction algorithm matching standard statistical libraries.
Beta distribution formulas
f(x) = x^(alpha-1) * (1-x)^(beta-1) / B(alpha, beta) for 0 <= x <= 1
B(alpha, beta) = gamma(alpha) * gamma(beta) / gamma(alpha + beta)
F(x) = I_x(alpha, beta), the regularised incomplete beta function
Mean = alpha / (alpha + beta)
Variance = alpha * beta / ((alpha + beta)^2 * (alpha + beta + 1))
Shape parameters above 1 give a single interior peak; values below 1 push mass toward the endpoints. Equal alpha and beta produce a distribution symmetric about 0.5.
Worked example and notes
- With alpha = 2, beta = 5, the mean is 2 / 7, about 0.2857.
- The variance is 10 / (49 times 8), about 0.0255.
- At x = 0.3 the density is about 2.1609.
- At x = 0.3 the cumulative probability is about 0.5798.
- alpha = beta = 1 gives the uniform distribution on [0, 1].
Beta distribution: frequently asked questions
What is the beta distribution?
The beta distribution is a continuous distribution on the interval from 0 to 1, defined by two positive shape parameters alpha and beta. Because its support is [0, 1] it is the natural model for probabilities and proportions, and it is the conjugate prior for the success probability of a binomial process.
What is the density of the beta distribution?
For x in [0, 1] the density is x to the power (alpha minus 1) times (1 minus x) to the power (beta minus 1), divided by the beta function B(alpha, beta). The beta function normalises the density so it integrates to 1, and equals gamma(alpha) times gamma(beta) over gamma(alpha plus beta).
How is the cumulative probability computed?
The cumulative distribution function is the regularised incomplete beta function I_x(alpha, beta). This calculator evaluates it with a continued-fraction method, the standard high-accuracy algorithm used in statistical software.
What are the mean and variance?
The mean is alpha divided by (alpha plus beta). The variance is alpha times beta divided by ((alpha plus beta) squared times (alpha plus beta plus 1)). Equal shapes give a symmetric distribution centred at 0.5.
What are valid inputs?
Both shape parameters alpha and beta must be strictly positive. The evaluation point x should lie in the interval from 0 to 1; values outside that range return a density of 0 and a cumulative value of 0 or 1.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Beta distribution.
- NIST Digital Library of Mathematical Functions: Incomplete beta functions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.