What this calculator does
The beta distribution is defined on the interval from 0 to 1, which makes it the natural distribution for a quantity that is itself a probability or a proportion. Its two shape parameters give it a remarkable range of forms.
The parameters have an intuitive reading in Bayesian work: α behaves like a count of successes and β like a count of failures. Beta(2,5) has a mean of 0.2857, roughly 2 out of 7, and leans left. Beta(1,1) is flat, the uniform distribution, representing no information at all. Beta(5,2) is the mirror image of Beta(2,5), with the same variance and a mean of 0.7143.
The formula
The density is x to the power α−1, times (1−x) to the power β−1, divided by the beta function B(α,β), which normalises the total area to 1. The mean is α/(α+β) and the variance is αβ divided by (α+β)²(α+β+1). Larger parameters concentrate the distribution more tightly around its mean.
| Term | Meaning |
|---|---|
| α (alpha) | First shape parameter, behaving like a success count. |
| β (beta) | Second shape parameter, behaving like a failure count. |
| Conjugate prior | Beta is the conjugate prior for a binomial proportion, so updating it with data keeps it beta. |
| Beta function | B(α,β), the normalising constant that makes the density integrate to 1. |
The inputs explained
| Field | What to enter |
|---|---|
| Shape α | Shape parameter α. Must be positive; it need not be a whole number. |
| Shape β | Shape parameter β. Must be positive. |
| x (0 to 1) | The point at which to evaluate the density, between 0 and 1. |
When to use it
Bayesian inference on a proportion
Starting from a beta prior and observing successes and failures gives a beta posterior, simply by adding the counts to the parameters.
Modelling a rate or share
Conversion rates, pass rates and any quantity confined to the unit interval.
Project time estimation
The PERT method uses a rescaled beta distribution to model task durations between an optimistic and a pessimistic bound.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the shape parameters change the distribution?
Several parameter combinations with their densities and moments.
| Shape α (β = 5) | f(x) (probability density) | Mean | Standard deviation |
|---|---|---|---|
| α = 0.5 | 0.108759 | 0.0909 | 0.1128 |
| α = 1 | 0.312500 | 0.1667 | 0.1409 |
| α = 2 | 0.937500 | 0.2857 | 0.1597 |
| α = 5 | 2.4609 | 0.5000 | 0.1508 |
Questions
Why is the beta distribution used for probabilities?
Because it is defined only on 0 to 1, which is exactly the range a probability can take, and its two parameters give enough flexibility to represent almost any belief about where in that range the true value lies.
What does Beta(1,1) mean?
It is the uniform distribution on 0 to 1, with a constant density of exactly 1 everywhere. As a prior it represents complete ignorance: every possible probability is equally plausible before seeing any data.
What is a conjugate prior?
A prior that produces a posterior in the same family. Starting with Beta(α,β) and observing s successes and f failures gives Beta(α+s, β+f) exactly. Updating requires only addition, which is why the beta-binomial pairing is so widely used.
Can alpha and beta be non-integers?
Yes, any positive values work. Beta(0.5, 0.5) is the Jeffreys prior, which is U-shaped with density piling up at both ends, expressing a belief that the true probability is likely near 0 or near 1 rather than in the middle.
For a proportion from sample data, see the sample proportion calculator. For updating beliefs with evidence, see the Bayes’ theorem calculator.