Beta Function Calculator

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

About Beta Function Calculator

The beta function is B(a, b) = Γ(a)Γ(b)/Γ(a + b), equal to the integral of x^(a−1)(1−x)^(b−1) from 0 to 1. It is symmetric in its arguments and underpins the beta distribution.

Formulas

Beta from gamma

B(a, b) = Γ(a)Γ(b) / Γ(a + b)

Whole-number case

B(m, n) = 1 / [(m + n − 1) · C(m + n − 2, m − 1)]

Worked example

With the default values below, Beta Function Calculator returns B(a, b) of 0.0095238095. Change any input above and every figure here is recalculated in your browser.

Inputs used

Parameter aminimum 0: 3
Parameter bminimum 0: 5

Results

B(a, b): 0.0095238095
1 / B(a, b): 105.00000000
Whole-number form 1/((a+b−1)·C(a+b−2, a−1)): 0.0095238095
B(b, a) — must match: 0.0095238095

Step by step

1Values used
2Beta from gamma
3Whole-number case
4B(a, b)= 0.0095238095
51 / B(a, b)= 105.00000000
6Whole-number form 1/((a+b−1)·C(a+b−2, a−1))= 0.0095238095
7B(b, a) — must match= 0.0095238095

Assumptions and limits

The result is derived only from the 2 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

What is B(3, 5)?

Γ(3)Γ(5)/Γ(8) = 2 × 24 / 5040 = 1/105 ≈ 0.0095238.

Why is it symmetric?

Swapping a and b just reflects the integrand about x = ½, which leaves the integral unchanged.