BETAINC

The regularized incomplete beta function evaluates a normalized cumulative integral of beta-kernel terms. It is widely used in cumulative probability calculations and ratio-based special-function identities.

It is defined as:

I_x(a,b)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_0^x t^{a-1}(1-t)^{b-1}\,dt

for positive a,b and 0\le x\le 1. This wrapper returns the regularized value in [0,1].

Excel Usage

=BETAINC(a, b, x)

a (float, required): First positive shape parameter.
b (float, required): Second positive shape parameter.
x (float, required): Upper integration bound in the unit interval.

Returns (float): Regularized incomplete beta value for the given parameters.

Example 1: Regularized incomplete beta at upper bound one

Inputs:

a	b	x
0.2	3.5	1

Excel formula:

=BETAINC(0.2, 3.5, 1)

Expected output:

1

Example 2: Regularized incomplete beta at midpoint

Inputs:

a	b	x
1.4	3.1	0.5

Excel formula:

=BETAINC(1.4, 3.1, 0.5)

Expected output:

0.81489

Example 3: Regularized incomplete beta with moderate shapes

Inputs:

a	b	x
2.2	3.1	0.4

Excel formula:

=BETAINC(2.2, 3.1, 0.4)

Expected output:

0.493396

Example 4: Regularized incomplete beta at lower bound zero

Inputs:

a	b	x
2	5	0

Excel formula:

=BETAINC(2, 5, 0)

Expected output:

0

Python Code

from scipy.special import betainc as scipy_betainc

def betainc(a, b, x):
    """
    Compute the regularized incomplete beta function.

    See: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betainc.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        a (float): First positive shape parameter.
        b (float): Second positive shape parameter.
        x (float): Upper integration bound in the unit interval.

    Returns:
        float: Regularized incomplete beta value for the given parameters.
    """
    try:
        if a <= 0 or b <= 0:
            return "Error: a and b must be positive"
        if x < 0 or x > 1:
            return "Error: x must be between 0 and 1"
        return float(scipy_betainc(a, b, x))
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

a *

First positive shape parameter.

b *

Second positive shape parameter.

x *

Upper integration bound in the unit interval.