Beta Distribution Calculator
This beta distribution calculator gives the density f(x), the cumulative probability P(X ≤ x) and P(X > x) for shape parameters α and β, plus mean, median, mode, variance, skewness and the quantile for a probability p.
Example: Shape parameter α (alpha) 2 · Shape parameter β (beta) 5 · Value x (between 0 and 1) 0.3 · Probability p for the quantile 0.95 → Result: P(X ≤ 0.3) = 0.579825 · density f(0.3) = 2.1609. Source: NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.17 Beta Distribution: f(x) = x^(p−1)(1−x)^(q−1)/B(p,q), F(x) = I_x(p,q), mean p/(p+q), mode (p−1)/(p+q−2), standard deviation, skewness (read 15.10.2026). Updated: .
How it is calculated
The beta distribution describes a share or probability between 0 and 1. Its two shape parameters α and β set where the mass lies: with α = 2 and β = 5 the mean is 2 ÷ 7 = 0.2857 and P(X ≤ 0.3) = 0.5798.
Formulas
- Density: f(x) = xα−1 (1 − x)β−1 ÷ B(α, β)
- CDF: F(x) = Ix(α, β), the regularized incomplete beta function
- Mean α ÷ (α + β), mode (α − 1) ÷ (α + β − 2) for α, β > 1
- Variance αβ ÷ ((α + β)² (α + β + 1))
For whole numbers there is a check via the binomial distribution: I0.5(2, 3) is the chance of at least 2 successes in 4 tries with p = 0.5, i.e. 11 ÷ 16 = 0.6875. You can verify this with the binomial distribution calculator.
Examples
| α | β | Mean | P(X ≤ 0.5) |
|---|---|---|---|
| 2 | 5 | 0.2857 | 0.8906 |
| 5 | 2 | 0.7143 | 0.1094 |
| 2 | 2 | 0.5 | 0.5 |
| 10 | 30 | 0.25 | 0.9995 |
Swapping α and β mirrors the distribution at 0.5. With α = β = 1 it is the uniform distribution; with both below 1 the density is U-shaped and largest at the edges.
Where it is used
As the prior and posterior for a success rate (Bayesian A/B tests: α = successes + 1, β = failures + 1), in PERT estimates and for proportions in general. Related models are the Weibull distribution for lifetimes and the normal distribution, which the beta distribution approaches for large α and β.
Frequently asked questions
How do I calculate the beta distribution CDF?
F(x) = I_x(α, β), the regularized incomplete beta function. It has no simple closed form; the calculator evaluates it with the continued fraction from NIST DLMF 8.17.22. Example: α = 2, β = 5 gives P(X ≤ 0.3) = 0.5798.
What is the mean of a beta distribution?
α ÷ (α + β). For α = 2 and β = 5 that is 0.2857.
What is the mode of the beta distribution?
(α − 1) ÷ (α + β − 2) if both parameters are greater than 1; for α = 2, β = 5 it is 0.2. With a parameter of 1 or less the maximum lies at the edge.
What does Beta(1, 1) mean?
It is the uniform distribution on [0, 1]: every value is equally likely, P(X ≤ x) = x.
How are α and β chosen from data?
For a success rate: α = successes + 1 and β = failures + 1 (uniform prior). 20 successes in 80 trials give Beta(21, 61) with mean 0.256.
Sources and legal basis
- NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.6.6.17 Beta Distribution: f(x) = x^(p−1)(1−x)^(q−1)/B(p,q), F(x) = I_x(p,q), mean p/(p+q), mode (p−1)/(p+q−2), standard deviation, skewness (read 15.10.2026)
- NIST Digital Library of Mathematical Functions §8.17(v), Eq. 8.17.22–8.17.23: continued fraction for the incomplete beta function I_x(a,b), symmetry 8.17.4 (read 15.10.2026)
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