Beta Distribution Calculator — PDF, CDF, Mean & Variance
The Beta distribution Beta(α, β) is a flexible continuous distribution on (0, 1) controlled by two positive shape parameters. It is widely used in Bayesian statistics as the conjugate prior for proportions, in PERT project scheduling, and for modelling any variable that must stay between 0 and 1. Enter α, β and a point x to get the PDF, CDF and summary statistics.
Probability that the Beta-distributed variable falls at or below x
- 1
α + β
2 + 5 = 7 - 2
Mean = α ÷ (α + β)
2 ÷ 7 = 0.2857 - 3
Variance denominator (α+β)² × (α+β+1)
7² × 8 = 392 - 4
Variance = α × β ÷ denominator
2 × 5 ÷ 392 = 0.0255Exact closed-form variance; the CDF above is computed by numerical integration.
How does this calculator work?
Beta(α, β) has PDF f(x) = x^(α−1)(1−x)^(β−1)/B(α,β) on (0,1), mean = α/(α+β), variance = αβ/((α+β)²(α+β+1)), mode = (α−1)/(α+β−2) when both > 1. The CDF P(X≤x) is evaluated by numerical integration. Set α and β to shape the distribution; enter x to read the probability and PDF value.
Formula
How this is calculated
The Beta distribution is shaped by two positive parameters α and β. When α = β = 1 the result is the uniform distribution. Both parameters above 1 give a unimodal bell; parameters below 1 concentrate mass at the boundaries; unequal parameters skew the curve left or right.
The PDF is evaluated using the Lanczos approximation to the log-gamma function to compute the normalising Beta function B(α, β) = Γ(α)Γ(β)/Γ(α+β). The CDF P(X ≤ x) — the regularised incomplete Beta function — is computed by 400-step trapezoidal integration of the PDF over [0, x], giving accuracy better than 5 × 10⁻⁴ for α, β ≥ 0.3.
The mean α/(α+β) and variance αβ/((α+β)²(α+β+1)) are exact. The mode (α−1)/(α+β−2) exists only when both α > 1 and β > 1. The PDF curve updates as you change the parameters so you can see the shape shift in real time.
Frequently asked questions
It models random variables constrained to (0, 1), making it ideal for probabilities and proportions. In Bayesian inference it is the conjugate prior for the Binomial likelihood, so the posterior is also Beta — a standard starting point for belief-updating. PERT project scheduling uses a scaled Beta to model uncertain task durations.
α > 1 pushes mass away from 0 (fewer very small values); β > 1 pushes mass away from 1. Equal parameters give a symmetric distribution centred at 0.5. Larger values together narrow the distribution; smaller values spread it toward the edges. α < 1 or β < 1 create a J-shape or U-shape.
The CDF uses 400-step trapezoidal integration of the PDF. For smooth combinations (α, β ≥ 0.3) accuracy is better than 5 × 10⁻⁴. For very small parameters (< 0.1) the PDF diverges near the boundaries and the trapezoidal method is less accurate; use a dedicated statistics library for high-precision work in those cases.
TG we-Calculate Editorial Team. (2026). Beta Distribution Calculator — PDF, CDF, Mean & Variance [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/beta-distribution-calculator
TG we-Calculate Editorial Team. "Beta Distribution Calculator — PDF, CDF, Mean & Variance." TG we-Calculate. 2026. https://we-calculate.com/calculator/beta-distribution-calculator.
TG we-Calculate Editorial Team, "Beta Distribution Calculator — PDF, CDF, Mean & Variance," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/beta-distribution-calculator
@misc{wecalculate_beta_distribution_calculator, title = {Beta Distribution Calculator — PDF, CDF, Mean & Variance}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/beta-distribution-calculator}}, year = {2026}, note = {TG we-Calculate} }
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