Beta Distribution Percent Point Calculator

Find the percent point (quantile) x of a beta distribution from a probability p and shape parameters alpha and beta, via the inverse regularized incomplete beta function. Lower and upper modes.

Input

Enter a probability p and the shape parameters alpha and beta to compute the percent point (quantile) x of a beta distribution.

Probability p

Enter a value between 0 and 1.

Mode

Choose whether p is the lower or the upper probability.

Shape parameter alpha

Enter a value greater than 0.

Shape parameter beta

Enter a value greater than 0.

Result

Percent point x for p = 0.95, alpha = 2, beta = 3

0.75139537

alpha / beta

0.66666667

Shape alpha

Shape beta

Lower probability F(x)

0.95

Upper probability 1 minus F(x)

0.05

Density f(x)

0.55727322

Mean

0.4

Variance

0.04

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

The cumulative distribution of a beta distribution is the regularized lower incomplete beta function F(x) = I_x(alpha, beta).
The percent point is the x that satisfies I_x(alpha, beta) = p. This tool uses bisection on the monotone increasing CDF to bracket the solution and then refines it with Newton iteration.
In upper mode the lower probability is converted to 1 minus p before solving.
The mean equals alpha divided by (alpha plus beta) and the variance equals alpha times beta divided by the product of (alpha plus beta) squared and (alpha plus beta plus 1).
lnGamma is evaluated with the Lanczos approximation and I_x with a continued fraction expansion.

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