Beta Distribution Calculator
Beta distribution with shape parameters α and β — cumulative probability P(X ≤ x), upper tail, density, mean, variance, mode and median for a value between 0 and 1.
How it works
f(x) = x^(α−1)(1 − x)^(β−1) ÷ B(α, β); P(X ≤ x) = I_x(α, β) (regularised incomplete beta); mean = α/(α + β); variance = αβ ÷ [(α + β)²(α + β + 1)]; mode = (α − 1)/(α + β − 2) for α, β > 1
The beta distribution lives on the interval 0 to 1, which makes it the natural model for proportions, probabilities and percentages: conversion rates, batting averages, task-completion shares. In Bayesian statistics it is the conjugate prior of the binomial — after s successes and f failures a Beta(α, β) prior becomes Beta(α + s, β + f). In project management (PERT) a scaled beta models task durations. Probabilities use the regularised incomplete beta function, as in R’s pbeta.
Worked example
For α = 2 and β = 5 (mean 0.2857), P(X ≤ 0.3) = 0.5798, P(X > 0.3) = 0.4202 and the density at 0.3 is 2.1609; the mode is 0.2000 and the median 0.2644.
Sources: NIST/SEMATECH e-Handbook — Beta distribution · SciPy — scipy.stats.beta
Frequently asked questions
What do α and β mean?
Think of them as prior successes and failures (plus one each): α pulls the mass towards 1, β towards 0. α = β = 1 is the flat uniform distribution.
How is it used in A/B testing?
Each variant’s conversion rate gets a Beta(1 + conversions, 1 + non-conversions) posterior; comparing those posteriors gives the probability one variant beats the other.
Why is the density above 1?
A density is not a probability. On an interval as short as 0–1 the density must exceed 1 somewhere unless the distribution is uniform; only areas under the curve are probabilities.
What happens when α or β is below 1?
The density shoots up towards 0 (α < 1) or 1 (β < 1), giving a U- or J-shaped curve, and the mode sits at the boundary.
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This calculator is for educational purposes. Double-check important results before acting on them.