Beta Distribution Calculator
Print PageA Beta Distribution Calculator (also known as a Beta PDF & CDF Calculator, Bayesian Beta Distribution Utility, Alpha & Beta Shape Parameter Calculator, or PERT Probability Density Analyzer) computes the Probability Density Function (PDF), Cumulative Distribution Function (CDF), expected mean (μ = α ÷ [α + β]), mode ([α - 1] ÷ [α + β - 2]), and variance for continuous random variables constrained within the bounded interval x ∈ [0, 1].
In Bayesian inference, machine learning, and conversion rate A/B testing, the Beta distribution serves as the conjugate prior for the Binomial distribution. When prior beliefs Beta(αprior, βprior) are combined with observed test data (k successes and n - k failures), the updated posterior distribution is simply Beta(α + k, β + n - k).
Our free online Beta Distribution Calculator provides instant calculations across all shape parameter combinations:
- Probability Density Function (PDF):
f(x; α, β) = [ xα-1 (1 - x)β-1 ] ÷ B(α, β). - Cumulative Distribution Function (CDF):
F(x; α, β) = Ix(α, β)(Regularized Incomplete Beta Function). - Expected Mean (μ):
μ = α ÷ (α + β). - Variance (σ2):
σ2 = (α · β) ÷ [ (α + β)2 · (α + β + 1) ]. - Distribution Mode (Peak Density for α,β > 1):
Mode = (α - 1) ÷ (α + β - 2). - Bayesian Posterior Updating Rule:
αpost = αprior + kandβpost = βprior + (n - k).
Master Beta Distribution Shape Parameter Reference Table
The table below displays the expected mean, variance, mode, distribution curve geometry, and real-world applications across standard α and β parameter combinations:
| Shape Parameters (α, β) | Curve Geometry Description | Expected Mean (μ) | Distribution Mode | Variance (σ2) | Primary Industry Application |
|---|---|---|---|---|---|
| α = 1, β = 1 | Flat Uniform Distribution [0, 1] | 0.5000 (50.0%) | Undefined (Flat) | 0.08333 | Uninformative Flat Bayes Prior |
| α = 21, β = 81 | Right-Skewed Narrow Bell Curve | 0.2059 (20.59%) | 0.2000 (20.00%) | 0.001587 (σ = 3.98%) | 20% Conversion A/B Test Posterior |
| α = 10, β = 10 | Symmetric Bell-Shaped Curve | 0.5000 (50.0%) | 0.5000 (50.00%) | 0.01190 (σ = 10.91%) | 50/50 Coin Toss Posterior |
| α = 0.5, β = 0.5 | U-Shaped Bimodal (Peaks at 0 and 1) | 0.5000 (50.0%) | 0.00 and 1.00 | 0.1250 | Jeffreys Prior for Bernoulli Trial |
| α = 2, β = 5 | Moderately Right-Skewed Curve | 0.2857 (28.57%) | 0.2000 (20.00%) | 0.02551 (σ = 15.97%) | Low Baseline Click-Through Rate |
Step-by-Step E-Commerce Conversion A/B Test Calculation
To evaluate a website landing page A/B test starting with an uninformative prior Beta(α=1, β=1), where k = 20 visitors convert out of n = 100 total visitors (80 non-conversions):
Step 1 (Update Posterior Shape Parameters): αpost = 1 + 20 = 21, and βpost = 1 + (100 - 20) = 81 &implies; Beta(21, 81)
Step 2 (Calculate Posterior Expected Mean μ): μ = 21 ÷ (21 + 81) = 21 ÷ 102 = 0.20588 ≈ 20.59%
Step 3 (Calculate Posterior Mode): Mode = (21 - 1) ÷ (21 + 81 - 2) = 20 ÷ 100 = 0.20000 = 20.00%
Step 4 (Calculate Posterior Variance σ^2): σ^2 = (21 · 81) ÷ [ 102^2 · 103 ] = 1,701 ÷ 1,071,612 = 0.0015873 &implies; σ = √0.0015873 ≈ 3.98%
Thus, the posterior distribution is centered at an expected conversion rate of 20.59% (Mode = 20.00%) with a standard error of 3.98%.
PERT Beta Distribution in Project Management
In project scheduling, the PERT (Program Evaluation and Review Technique) uses a specialized Beta distribution to model task duration based on three estimations:
| PERT Estimate Metric | Mathematical Formula | Example Calculation (a=2d, m=5d, b=14d) |
|---|---|---|
| PERT Expected Duration (μ) | (a + 4m + b) ÷ 6 | (2 + 4[5] + 14) ÷ 6 = 36 ÷ 6 = 6.0 Days |
| PERT Standard Deviation (σ) | (b – a) ÷ 6 | (14 – 2) ÷ 6 = 12 ÷ 6 = 2.0 Days |
History & Mathematics: 1763 Thomas Bayes to 1895 Karl Pearson
1763 Thomas Bayes & Conjugate Priors
In 1763, minister and mathematician Thomas Bayes (published posthumously by Richard Price) formulated Bayes’ Theorem in Philosophical Transactions of the Royal Society, establishing the foundation for continuous prior distributions on unknown probabilities.
1895 Karl Pearson & Pearson Type I Distribution
In 1895, English mathematician Karl Pearson formally introduced the Beta distribution as the Pearson Type I distribution in his systematic classification of continuous probability curves.
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Frequently Asked Questions (FAQ)
What is the Beta distribution used for?
The Beta distribution models continuous probabilities constrained between 0 and 1 (or 0% and 100%), widely used in Bayesian A/B testing and project management.
What is the formula for the mean of a Beta distribution?
The mean is calculated as μ = α ÷ (α + β).
Why is the Beta distribution conjugate to the Binomial distribution?
Because multiplying a Beta prior by a Binomial likelihood results in a posterior distribution that is also a Beta distribution with parameters α + k and β + n - k.