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]. Flat, no mode at all.
• alpha = beta > 1: symmetric and bell-shaped, centred at 0.5.
• alpha > beta: skewed left, mass concentrated toward 1.
• alpha < beta: skewed right, mass concentrated toward 0.
• alpha < 1 or beta < 1: U-shaped or J-shaped, 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 formula (alpha - 1) / (alpha + beta - 2) only applies when both parameters exceed 1. Outside that case the peak sits on a boundary rather than inside the interval, and for Beta(1, 1) there is no peak because the density never changes. Plenty of references print the formula without the condition, which is how you end up seeing a “mode” of 0.5 quoted for a flat distribution.

Worked example: Beta(2, 5) at x = 0.3

With alpha = 2 and beta = 5 the density has a closed form, f(x) = 30x(1-x)^4, so this one can be checked by hand:

f(0.3) = 30 × 0.3 × 0.7^4 = 9 × 0.2401 = 2.1609 The CDF integrates to F(x) = 1 - (1-x)^5 (1 + 5x), giving F(0.3) = 1 - 0.16807 × 2.5 = 0.579825 Mean = 2/7 = 0.2857, mode = (2-1)/(2+5-2) = 0.2

So a little under 58% of the distribution sits below 0.3, and the density there is 2.16. A density above 1 is normal and is not a probability: the area under the whole curve is 1, but the curve itself can be as tall as it likes.

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.