Inverse Regularized Incomplete Beta Function Calculator

Solve I_x(a,b)=p for x using Newton and bisection methods. Handy for beta distribution quantiles.

Input

Find x such that the regularized incomplete beta function I_x(a,b)=p. The value x is the quantile of the beta distribution.

Probability p

Between 0 and 1

Shape parameter a

Positive real number

Shape parameter b

Positive real number

Solution x

0.3857275681

x such that I_x(2,3) = 0.5

Check I_x(a,b)

0.5

Residual |I_x(a,b) − p|

1.665335e-16

Method

Newton

Iterations

Curve of I_x(a,b)

Plots I_x(a,b) against x and highlights the solution point.

The regularized incomplete beta function I_x(a,b) is evaluated with a continued fraction (Lentz algorithm), and lnΓ uses the Lanczos approximation.
The inverse is refined from an analytic initial guess with Newton iteration, falling back to bisection when convergence is insufficient.
The derivative dI/dx uses the probability density x^(a-1)(1-x)^(b-1)/B(a,b).
For verification, I_x(a,b) is recomputed at the solved x and the residual against the input p is shown.
Enter p between 0 and 1, with shape parameters a and b both positive real numbers.

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