Beta Distribution Calculator

Work out beta distribution instantly with clear inputs, formula shown and shareable results.

About Beta Distribution Calculator

The beta distribution lives on the interval from 0 to 1, which makes it the natural model for a probability or proportion. It is the conjugate prior for the binomial, so α and β act as counts of prior successes and failures.

Formulas

Beta density

f(x) = x^(α−1)(1−x)^(β−1) / B(α, β)

Mean and mode

mean = α/(α+β), mode = (α−1)/(α+β−2) for α, β > 1

Worked example

With the default values below, Beta Distribution Calculator returns P(X ≤ x) of 0.5798250000. Change any input above and every figure here is recalculated in your browser.

Inputs used

Shape αminimum 0: 2
Shape βminimum 0: 5
Evaluate at x (between 0 and 1)accepts 0–1: 0.3

Results

P(X ≤ x): 0.5798250000
Probability density at x: 2.1609000000
Mean α/(α+β): 0.28571429
Mode (α−1)/(α+β−2): 0.20000000

Step by step

1Values used
2Beta density
3Mean and mode
4P(X ≤ x)= 0.5798250000
5Probability density at x= 2.1609000000
6Mean α/(α+β)= 0.28571429
7Mode (α−1)/(α+β−2)= 0.20000000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For α = 2, β = 5 at x = 0.3?

P(X ≤ 0.3) ≈ 0.5798, with a mean of 2/7 ≈ 0.2857 and a mode of 0.2.

Why is it the binomial's conjugate prior?

Multiplying a beta prior by a binomial likelihood gives another beta, with the successes and failures simply added to α and β.