The beta distribution is the natural language for proportions: it lives entirely between 0 and 1 and bends into almost any shape — bell, U, J, or flat — as its two parameters change. This Beta Distribution Calculator takes the shape parameters α and β plus a point x, and returns the density (PDF), the cumulative probability (CDF), the mean, variance and mode, so you can interrogate any beta model numerically instead of squinting at sketches.
Worked Example: Beta(2, 5) at x = 0.3
For α = 2 and β = 5 the calculator reports a mean of 0.2857 (= 2/7), a variance of 0.025510, a density at 0.3 of 2.1609, and a cumulative probability P(X ≤ 0.3) of 0.5798. Each number can be checked independently. The density: B(2, 5) = Γ(2)Γ(5)/Γ(7) = (1! × 4!)/6! = 1/30, so f(0.3) = 0.3¹ × 0.7⁴ / (1/30) = 30 × 0.3 × 0.2401 = 2.1609. The CDF has a binomial identity for integer shapes — I₀.₃(2, 5) equals the probability that a Binomial(6, 0.3) variable is at least 2, which sums to 0.5798. And the CDF passing 0.5 below x = 0.3 is consistent with a mean of 0.2857 and a right-skewed shape: most of the mass sits in the left third, with a long tail reaching toward 1 — the classic profile of a conversion rate or a proportion that is usually small but occasionally large.
The Formulas
The density is f(x) = x^(α−1) (1−x)^(β−1) / B(α, β), where B(α, β) = Γ(α)Γ(β)/Γ(α+β) is the beta function — a normalizing constant whose only job is making the density integrate to 1. The mean is α/(α+β); the variance is αβ / [(α+β)²(α+β+1)]; the mode, when both parameters exceed 1, is (α−1)/(α+β−2). The CDF is the regularized incomplete beta function I_x(α, β), which has no closed form in general — this calculator evaluates it with the standard continued-fraction expansion, the same algorithm statistical software uses. Special cases anchor intuition: α = β = 1 is the uniform distribution on [0, 1]; α = β = 0.5 is the arcsine distribution, piling mass at the extremes; large equal α = β gives a symmetric bell tightening around 0.5. Reading any beta model starts from the mean formula: α/(α+β) is where the distribution centers, and α+β acts as a concentration dial — bigger sums, tighter distribution.
Where the Beta Distribution Works
Bayesian statistics made it famous: the beta is the conjugate prior for a binomial proportion, so a Beta(α, β) prior updated with s successes and f failures becomes Beta(α+s, β+f) — the posterior mean (α+s)/(α+β+s+f) is just this page’s mean formula with the data folded in, which is why A/B testing tools and conversion dashboards quietly run on beta arithmetic. Project management: the PERT three-point estimate is a beta distribution fitted to optimistic, likely and pessimistic durations. Reliability and quality: proportions of defective items, machine yields and click-through rates all model naturally on [0, 1]. Order statistics: the k-th smallest of n uniform samples is exactly beta-distributed. In each case the calculator’s job is the same: turn a stated (α, β) model into probabilities of intervals — P(X ≤ x) here, and P(a ≤ X ≤ b) by subtracting two CDF values.
Common Mistakes
Reading the PDF as a probability: a density of 2.1609 is not a 216% chance — densities are heights, and only areas (CDF differences) are probabilities; on a [0, 1] support, heights above 1 are routine. Feeding x outside [0, 1], where the distribution simply does not live. Confusing the parameters’ roles: swapping α and β mirrors the distribution around 0.5. And treating the mean as the whole story for U-shaped parameter choices (α, β below 1), where the mean sits at the least likely value in the middle — check the mode line and the CDF shape before summarizing.
Reading α + β as a Prior Sample Size
The sum α + β behaves like an effective sample size behind the prior: Beta(2, 5) carries the weight of about 7 previous observations (2 successes, 5 failures), while Beta(20, 50) carries the same mean with ten times the conviction. The update arithmetic shows the consequence. Start from Beta(2, 5) and observe 30 trials with 12 successes: the posterior is Beta(14, 23), whose mean — this calculator’s formula — is 14/37 ≈ 0.378, pulled from the prior mean 0.286 toward the observed rate 0.40, but not all the way. Start instead from the flatter Beta(1, 1) and the same data lands at a mean of 13/32 ≈ 0.406, nearly the raw rate. Same evidence, different prior weight — the calculator’s mean and CDF lines let you price exactly how much any stated prior is steering the answer.
Related Tools
Continue with the Lognormal Distribution Calculator, the Weibull Distribution Calculator, the False Positive Calculator, or browse the full Statistics Calculators hub.
Frequently Asked Questions
What is the beta distribution used for?
Modeling quantities confined to [0, 1]: proportions, rates, probabilities themselves. It is the standard Bayesian prior for a binomial success rate and the shape behind PERT estimates and A/B test posteriors.
What are the mean and variance of Beta(2, 5)?
Mean = α/(α+β) = 2/7 ≈ 0.2857. Variance = αβ/[(α+β)²(α+β+1)] = 10/392 ≈ 0.025510, so the standard deviation is about 0.1597.
How do I get P(0.2 ≤ X ≤ 0.4) from this calculator?
Run it twice: CDF at 0.4 minus CDF at 0.2. Interval probabilities are always differences of cumulative probabilities.
Why can the PDF be bigger than 1?
Because a density is not a probability — it is probability per unit of x. On a support only 1 unit wide, a concentrated distribution must rise above 1 somewhere for its total area to equal 1.
What does α = β = 1 give?
The uniform distribution on [0, 1]: density 1 everywhere, CDF(x) = x, mean 0.5. It is the beta family’s flat special case.
How does the beta update with data?
As a prior for a proportion, observing s successes and f failures turns Beta(α, β) into Beta(α + s, β + f). The parameters behave like prior pseudo-counts of each outcome.