Beta Distribution Calculator

About the Beta Distribution Calculator

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 Jeffreys' prior (0.5, 0.5). Use the posterior update to confirm that adding successes shifts mass toward 1 and adding failures shifts it toward 0.

When This Page Helps

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.

How to Use the Inputs

1. Enter the shape parameter α (alpha) — values > 1 concentrate mass away from 0.
2. Enter the shape parameter β (beta) — values > 1 concentrate mass away from 1.
3. Set x for point evaluation of the PDF and CDF.
4. Set x₁ and x₂ for interval probability P(x₁ ≤ X ≤ x₂).
5. Use preset buttons to explore common distribution shapes.
6. Read mean, mode, variance, skewness and other statistics from the output.
7. Review the density curve and percentile table for a complete picture.

Example Calculation

Tips & Best Practices

• α = β = 1 gives the uniform distribution on [0,1] — all values equally likely.
• When both α and β are less than 1, the distribution is U-shaped (mass at extremes).
• The mean is always α/(α+β), so α = 2, β = 8 gives a mean of 0.2 — useful for modeling rare event probabilities.
• In Bayesian A/B testing, start with Beta(1,1) as a non-informative prior and update with observed conversions.
• Larger α + β (while keeping ratio fixed) means a more concentrated distribution — less uncertainty.
• The beta distribution generalizes to a 4-parameter version with arbitrary [a,b] bounds.

Understanding Shape Parameters

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.

Bayesian A/B Testing with Beta Distributions

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.

Relationship to Other Distributions

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.

Frequently Asked Questions

• What do α and β represent?

  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.

• Why can the PDF exceed 1?

  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).

• How is the beta distribution used in Bayesian statistics?

  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.

• What's a non-informative prior for the beta distribution?

  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.

• Can I use this for modeling batting averages?

  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.

• How does the beta distribution relate to the binomial?

  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.