Incomplete Beta Function Calculator

Enter x, a, b to compute the regularized incomplete beta function I_x(a,b) and B(x;a,b) via continued fractions.

Input

Enter x (0 to 1) and the shape parameters a and b to compute the regularized incomplete beta function I_x(a,b) via a continued fraction.

Upper limit x

Between 0 and 1

Parameter a

Positive real number

Parameter b

Positive real number

Regularized incomplete beta I_x(a,b)

0.6875

x = 0.5, a = 2, b = 3

B(x;a,b) / B(a,b)

0.6875

Incomplete beta B(x;a,b)

0.05729167

Complete beta B(a,b)

0.08333333

Curve of I_x(a,b)

I_x(a,b) as x moves from 0 to 1 with a and b fixed. The dot marks the entered x.

Computes the regularized incomplete beta function I_x(a,b) = B(x;a,b) / B(a,b), where B(x;a,b) is the incomplete beta function and B(a,b) is the complete beta function.
The continued fraction is evaluated with the modified Lentz method. When x exceeds (a+1)/(a+b+2) convergence is poor, so the symmetry relation I_x(a,b) = 1 minus I_(1-x)(b,a) swaps the arguments.
The complete beta function B(a,b) = gamma(a)gamma(b)/gamma(a+b) is obtained from ln gamma via the Lanczos approximation.
Valid input ranges are x between 0 and 1 inclusive, with a and b positive real numbers. Out-of-range input shows an error.
The chart sweeps x from 0 to 1 for the current a and b, plotting the I_x(a,b) curve and marking the point at the entered x.

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