Beta Distribution Calculator

Compute the beta distribution PDF, CDF (lower probability), upper probability, mean, and variance from shape parameters α, β and x, with charts.

Input

Enter the shape parameters α and β and a value x between 0 and 1 to compute the beta distribution density, cumulative probability, upper probability, mean, and variance.

Value x (0 to 1)

Enter a real number between 0 and 1.

Shape parameter α

Enter a real number greater than 0.

Shape parameter β

Enter a real number greater than 0.

Result

Cumulative distribution F(x) for x = 0.5, α = 2, β = 5

0.890625

Density f(x)

0.9375

Upper probability 1 − F(x)

0.109375

Mean

0.28571429

Variance

0.0255102

Probability density function PDF

Cumulative distribution function CDF

How it works

The density is f(x) = x^(α−1)(1−x)^(β−1) / B(α, β) for 0 ≤ x ≤ 1, where B(α, β) is the complete beta function.
The cumulative distribution (lower probability) F(x) is the regularized incomplete beta function I_x(α, β). The upper probability is 1 − F(x).
The mean is α/(α+β) and the variance is αβ/((α+β)²(α+β+1)).
The regularized incomplete beta function is evaluated with a Lentz continued fraction, swapping arguments via the symmetry I_x(α,β) = 1 − I_(1−x)(β,α) when it improves convergence.
lnΓ is computed with the Lanczos approximation. Extreme parameters may introduce numerical error.

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