Beta Distribution Calculator (PDF, CDF, mean, mode, variance)

Understanding the beta distribution calculator — concept and principle

This calculator gives the cumulative probability, the density and the summary statistics of a beta distribution, the standard model for a quantity between 0 and 1 such as a proportion, a success rate or a probability. Use it in Bayesian statistics, project estimates (PERT) and reliability work.

How it is calculated

Density f(x) = x^(α−1) (1 − x)^(β−1) ÷ B(α, β), for 0 ≤ x ≤ 1

B(α, β) = Γ(α) Γ(β) ÷ Γ(α + β)

P(X ≤ x) = Iₓ(α, β), the regularized incomplete beta function

Mean = α ÷ (α + β); variance = αβ ÷ [(α + β)² (α + β + 1)]

Mode = (α − 1) ÷ (α + β − 2) when α, β > 1

Skewness = 2(β − α) √(α + β + 1) ÷ [(α + β + 2) √(αβ)]

• The median has no closed form and is found numerically from Iₓ(α, β) = 0.5.

The principle

The two shape parameters act like counts of successes and failures: α pulls the mass toward 1 and β toward 0. In Bayesian updating, a beta prior combined with binomial data gives another beta distribution, with α increased by the successes and β by the failures. α = β = 1 is the uniform distribution; α = β > 1 is symmetric around 0.5; and larger α + β makes the distribution narrower, like having more data.

Worked example

Default inputs: α = 2, β = 5, x = 0.3.

1. B(2, 5) = 1! × 4! ÷ 6! = 1/30, so f(0.3) = 30 × 0.3 × 0.7⁴ = 2.1609.
2. For whole-number parameters, P(X ≤ 0.3) = P(at least 2 successes in 6 trials with p = 0.3) = 1 − 0.7⁶ − 6 × 0.3 × 0.7⁵ = 0.579825; P(X > 0.3) = 0.420175.
3. Mean = 2 ÷ 7 = 0.285714; mode = 1 ÷ 5 = 0.2; median = 0.26445.
4. Variance = 10 ÷ (49 × 8) = 0.0255102; standard deviation = 0.159719.
5. Skewness = 2 × 3 × √8 ÷ (9 × √10) = 0.596285.

How to read the result

Here the distribution is skewed right: most mass sits around 0.2–0.3 with a tail toward higher values, so mode < median < mean. A density above 1 is normal for continuous distributions on a short interval; only the area under the curve must equal 1.

Common mistakes

• Reading the density as a probability.
• Swapping α and β, which mirrors the distribution around 0.5.
• Expecting a mode when α or β is 1 or less; the peak is then at an end.
• Using x outside 0–1.

Frequently asked questions

What is the beta distribution used for?

Modelling proportions and probabilities, especially as a prior in Bayesian statistics.

What does Beta(1, 1) look like?

It is the uniform distribution on 0 to 1.

How do I update a beta prior?

Add the number of successes to α and the number of failures to β.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.17 Beta Distribution