About

The Beta distribution is defined on the interval [0, 1] and parameterized by two shape parameters α and β. It models proportions, probabilities, and bounded random variables in Bayesian inference, A/B testing, and reliability engineering. Misconfiguring shape parameters leads to incorrect posterior estimates, faulty credible intervals, and flawed hypothesis tests. This calculator uses the Lanczos approximation for the log-gamma function and Lentz’s continued fraction algorithm for the regularized incomplete beta function I_x(α, β), matching reference implementations to 10⁻¹⁰ relative accuracy.

The tool computes the full moment profile: mean, variance, standard deviation, skewness, excess kurtosis, and mode (when defined). It also evaluates the PDF f(x), CDF F(x), and inverse CDF (quantile) via Newton-Raphson iteration. Note: the mode is undefined when both α ≤ 1 and β ≤ 1 (the U-shaped case). For extreme parameter ratios (α/β > 10000), numerical precision may degrade near boundary values.

Formulas

The probability density function (PDF) of the Beta distribution is:

f(x; α, β) = x^{α − 1} ⋅ (1 − x)^{β − 1}B(α, β)

where B(α, β) is the Beta function:

B(α, β) = Γ(α) ⋅ Γ(β)Γ(α + β)

The cumulative distribution function (CDF) is the regularized incomplete beta function:

F(x; α, β) = I_x(α, β) = B(x; α, β)B(α, β)

Key moments and properties:

Mean: μ = αα + β

Variance: σ² = α ⋅ β(α + β)² ⋅ (α + β + 1)

Mode: α − 1α + β − 2 for α > 1, β > 1

Skewness: 2(β − α) ⋅ √α + β + 1(α + β + 2) ⋅ √α ⋅ β

Kurtosis (excess): 6[(α − β)²(α + β + 1) − αβ(α + β + 2)]αβ(α + β + 2)(α + β + 3)

Where α = first shape parameter (alpha), β = second shape parameter (beta), x = evaluation point in [0, 1], Γ = Gamma function, I_x = regularized incomplete beta function computed via continued fraction expansion.

Reference Data

Shape Profile	α	β	Mean	Variance	Mode	Skewness	Description
Uniform	1	1	0.5000	0.0833	Any	0	Flat distribution, maximum entropy on [0,1]
Symmetric bell	5	5	0.5000	0.0227	0.5000	0	Symmetric, concentrated at center
Right-skewed	2	5	0.2857	0.0255	0.2000	0.5963	Mass shifted toward 0
Left-skewed	5	2	0.7143	0.0255	0.8000	−0.5963	Mass shifted toward 1
U-shaped	0.5	0.5	0.5000	0.1250	Bimodal	0	Arcsine distribution, mass at boundaries
J-shaped (left)	0.5	1	0.3333	0.0889	0	0.5657	Monotone decreasing
J-shaped (right)	1	0.5	0.6667	0.0889	1	−0.5657	Monotone increasing
Peaked center	10	10	0.5000	0.0119	0.5000	0	Tight bell, low variance
Jeffreys prior	0.5	0.5	0.5000	0.1250	Bimodal	0	Non-informative Bayesian prior for binomial
Haldane prior	0.01	0.01	0.5000	0.2475	Bimodal	0	Improper prior, extreme boundary mass
Bayesian posterior (weak)	2	2	0.5000	0.0500	0.5000	0	After 1 success, 1 failure with uniform prior
A/B test (100 vs 90)	101	901	0.1008	0.0001	0.1000	0.0794	Conversion rate posterior after 1000 trials
Strong left evidence	1	10	0.0909	0.0069	0	1.0646	Heavy right skew, near-zero proportion
Strong right evidence	10	1	0.9091	0.0069	1	−1.0646	Heavy left skew, near-one proportion
Exponential-like	1	5	0.1667	0.0198	0	0.8528	Monotone decreasing from boundary
High concentration	50	50	0.5000	0.0025	0.5000	0	Very tight, quasi-normal
Power law	0.3	2	0.1304	0.0343	0	1.5811	Heavy-tailed toward zero
Quasi-Bernoulli	0.1	0.1	0.5000	0.2083	Bimodal	0	Almost all mass at 0 and 1

Frequently Asked Questions

When both α and β exceed 1, the distribution is unimodal with a bell shape. When both are less than 1, it becomes U-shaped (bimodal at the boundaries). If α < 1 and β ≥ 1, the density is J-shaped with infinite density at x = 0. The ratio α/β controls skewness: α > β shifts mass toward 1, while α < β shifts mass toward 0. Equal values (α = β) produce symmetric distributions centered at 0.5.

The mode formula (α − 1)/(α + β − 2) is only valid when both α > 1 and β > 1. When α = β = 1 (uniform distribution), every point is equally likely. When α < 1 or β < 1, the density diverges at the boundaries (0 or 1), making the mode a boundary value rather than an interior maximum. For α ≤ 1 and β ≤ 1 simultaneously, the distribution is U-shaped with two boundary modes.

The calculator uses Lentz's modified continued fraction algorithm applied to the regularized incomplete beta function I_x(α, β). For values where x > (α + 1)/(α + β + 2), it applies the symmetry relation I_x(α, β) = 1 − I_{1−x}(β, α) to ensure the continued fraction converges rapidly. The log-gamma function uses the Lanczos approximation with g = 7 and standard coefficients, providing at least 10 significant digits of accuracy.

The quantile function uses Newton-Raphson iteration. Given a target probability p, it finds x such that I_x(α, β) = p. The initial guess uses a rational approximation based on the normal quantile transformed through the Wilson-Hilferty cube root approximation. Each iteration updates x by subtracting (CDF(x) − p) / PDF(x), with bounds clamping to [0, 1]. Convergence typically occurs within 10-15 iterations to machine precision.

Yes. All computations use log-space arithmetic (log-gamma, log-PDF) to avoid overflow. The Beta function B(α, β) for large parameters would underflow in direct computation, but ln B(α, β) = ln Γ(α) + ln Γ(β) − ln Γ(α + β) remains numerically stable. The distribution plot adapts its x-axis range to focus on the region where the density is non-negligible, which for large equal parameters is a narrow band around 0.5.

In Bayesian analysis with a binomial likelihood, the Beta distribution is the conjugate prior. Starting with a prior Beta(α₀, β₀), after observing s successes and f failures, the posterior is Beta(α₀ + s, β₀ + f). For A/B testing, each variant's conversion rate is modeled as a Beta posterior. The probability that variant A beats variant B is computed by comparing their posterior distributions. Common non-informative priors include the uniform Beta(1, 1), the Jeffreys prior Beta(0.5, 0.5), and the Haldane prior Beta(ε, ε) where ε → 0.

Beta(1, 1) is the continuous uniform on [0, 1]. Beta(0.5, 0.5) is the arcsine distribution. If X ~ Beta(α, β), then (1 − X) ~ Beta(β, α). The Beta distribution is related to the F-distribution: if X ~ Beta(α, β), then X/(1−X) · β/α ~ F(2α, 2β). For large α + β with fixed mean, Beta converges to a Normal distribution. The order statistics of n uniform samples follow Beta distributions: the k-th smallest is Beta(k, n − k + 1).