Beta Distribution Calculator

The beta distribution is the natural home for probabilities, proportions, and rates — quantities that must fall between 0 and 1 and cannot be negative or exceed unity. Where the normal distribution models unbounded quantities and the Poisson models count data, the beta distribution models the probability that a probability takes a particular value. This self-referential elegance makes it the foundation of Bayesian inference, where it serves as the conjugate prior for Bernoulli and binomial likelihoods. The beta distribution calculator computes all key distributional statistics from your shape parameters.

The Beta Distribution: Shape Parameters and Their Meaning

The beta distribution Beta(α, β) is defined on the interval [0, 1] with two positive shape parameters:

• α (alpha): counts of "successes" + 1 in the Bayesian updating interpretation; increases α shifts the distribution rightward (toward higher probabilities)
• β (beta): counts of "failures" + 1 in Bayesian updating; increases β shifts the distribution leftward

Special cases with intuitive interpretations:

• Beta(1, 1) = Uniform distribution — complete prior ignorance; all values equally likely
• Beta(α, α) = symmetric distribution centered at 0.5; higher α = more peaked
• Beta(α > 1, β > 1) = unimodal (single peak); mode = (α − 1)/(α + β − 2)
• Beta(α < 1, β < 1) = U-shaped bimodal; probability mass concentrated near 0 and 1
• Beta(α > 1, β = 1) = strictly increasing; Beta(α = 1, β > 1) = strictly decreasing

Use this online calculator for any α and β combination. The Bayesian updating calculator applies the beta distribution's conjugate prior property to real inference problems.

Key Statistics: Mean, Variance, and Mode

The analytical formulas for beta distribution moments:

• Mean: μ = α / (α + β) — always between 0 and 1; equals the proportion estimate after observing α − 1 successes in α + β − 2 trials
• Variance: σ² = αβ / [(α + β)²(α + β + 1)] — decreases as total count (α + β) increases, reflecting more certainty with more data
• Mode: (α − 1) / (α + β − 2) for α > 1 and β > 1 — the maximum likelihood estimate (sample proportion) for Bayesian point estimation
• Standard deviation: σ = √(αβ / [(α+β)²(α+β+1)])

Bayesian A/B Testing: Beta Distribution in Action

The most accessible modern application of the beta distribution is Bayesian A/B testing. Suppose a website control page converts at an unknown rate p_A. After observing 45 conversions in 200 visits, the posterior distribution for p_A is Beta(α = 46, β = 156) — with the prior Beta(1, 1) representing no prior knowledge. The posterior mean = 46/202 = 22.8%; the 95% credible interval bounds the true rate with 95% probability (unlike a frequentist confidence interval). To assess which variant wins, compare the posterior distributions of p_A and p_B directly: P(p_B > p_A) = ∫ across all possible combinations — a calculation that requires numerical integration but is conceptually transparent. The probability calculator and probability distribution calculators cover the complete statistical toolkit.

PERT Distribution: Beta in Project Management

The Program Evaluation and Review Technique (PERT) uses a modified beta distribution to model task duration uncertainty. Given optimistic (a), most likely (m), and pessimistic (b) estimates: PERT mean = (a + 4m + b) / 6; PERT standard deviation = (b − a) / 6. These parameters map to the beta distribution's α and β through the PERT equations: α = 6 × (mean − a) / (b − a); β = 6 × (b − mean) / (b − a). The PERT distribution is the project manager's practical tool for converting three-point expert estimates into a probabilistic duration model without requiring full distributional data.