Compute PDF, CDF, mean, mode, variance, and percentiles for the beta distribution with interactive shape parameter presets and visual curve.
The beta distribution calculator computes probability density, cumulative distribution, and key statistics for the Beta(α, β) distribution. The beta distribution is defined on the interval [0, 1] and is incredibly flexible — it can model uniform, skewed, U-shaped, or bell-shaped distributions depending on the shape parameters α and β.
This distribution is the conjugate prior for the binomial likelihood in Bayesian statistics, making it fundamental for Bayesian inference about proportions and probabilities. If you observe k successes in n trials and start with a Beta(α₀, β₀) prior, the posterior is Beta(α₀ + k, β₀ + n − k).
The calculator provides point PDF and CDF evaluation, interval probabilities, a visual density curve, and a percentile table. Preset configurations let you quickly explore common shapes: uniform (1,1), right-skewed (2,5), left-skewed (5,2), and the Jeffrey's prior (0.5, 0.5). Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
The beta distribution appears throughout Bayesian statistics, reliability engineering, project management (PERT estimation), and A/B testing. Its flexibility on the [0, 1] interval makes it the natural choice for modeling probabilities, proportions, and rates.
This calculator makes it easy to explore different parameterizations, compute exact probabilities, and visualize the distribution shape — all essential for statistical modeling.
f(x; α, β) = x^(α−1) (1−x)^(β−1) / B(α, β), where B(α, β) = Γ(α)Γ(β)/Γ(α+β). Mean = α/(α+β). Mode = (α−1)/(α+β−2) for α,β > 1. Variance = αβ/[(α+β)²(α+β+1)].
Result: f(0.7) ≈ 2.0412, F(0.7) ≈ 0.5798
For Beta(5, 2), the PDF at x = 0.7 is about 2.04 (density can exceed 1) and the CDF is about 0.58, meaning there's a 58% probability that X ≤ 0.7.
When α = β, the distribution is symmetric around 0.5. Increasing both while keeping them equal makes the distribution taller and narrower — more concentrated around 0.5. When α > β, the distribution skews left (mass toward 1); when α < β, it skews right (mass toward 0). The special case α = β = 1 is the uniform distribution.
In A/B testing, each variant's conversion rate is modeled as a beta distribution. Start with Beta(1, 1) for each. After observing conversions and non-conversions, update to Beta(1 + conversions, 1 + non-conversions). The probability that variant A beats B is computed by sampling or numerical integration.
The beta distribution is related to many other distributions: the uniform is Beta(1,1), the arcsine is Beta(0.5, 0.5), and the beta-binomial is obtained by mixing a binomial with a beta prior. It's also connected to the F-distribution through a simple transformation.
They're shape parameters. In Bayesian terms, α − 1 can be thought of as "prior successes" and β − 1 as "prior failures." Larger values mean more prior information and a tighter distribution.
The PDF is a density, not a probability. It can exceed 1 at any point as long as the total area under the curve equals 1. This commonly happens with concentrated distributions (large α, β).
It's the conjugate prior for binomial data. If your prior is Beta(α, β) and you observe k successes in n trials, the posterior is Beta(α + k, β + n − k). This makes updating beliefs analytically tractable.
Beta(1, 1) is uniform and treats all probabilities as equally likely. Beta(0.5, 0.5) is Jeffrey's prior, which is invariant under reparameterization. Both are commonly used non-informative priors.
Yes! Batting averages are bounded between 0 and 1 and can be well-modeled by a beta distribution. The empirical Bayes approach fits α and β from historical data to shrink extreme averages toward the mean.
If X ~ Binomial(n, p) and p ~ Beta(α, β), then the posterior distribution of p given X = k is Beta(α + k, β + n − k). The beta and binomial form a conjugate pair.