scipy.special.betainc#

scipy.special.betainc(a, b, x, out=None) = <ufunc 'betainc'>#

Regularized incomplete beta function.

Computes the regularized incomplete beta function, defined as [1]:

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

for \(0 \leq x \leq 1\).

Parameters:

a, barray_like: Positive, real-valued parameters
xarray_like: Real-valued such that \(0 \leq x \leq 1\), the upper limit of integration
outndarray, optional: Optional output array for the function values

Returns:

scalar or ndarray: Value of the regularized incomplete beta function

See also

beta: beta function
betaincinv: inverse of the regularized incomplete beta function

Notes

The term regularized in the name of this function refers to the scaling of the function by the gamma function terms shown in the formula. When not qualified as regularized, the name incomplete beta function often refers to just the integral expression, without the gamma terms. One can use the function beta from scipy.special to get this “nonregularized” incomplete beta function by multiplying the result of betainc(a, b, x) by beta(a, b).

References

[1]

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.17

Examples

Let \(B(a, b)\) be the beta function.

>>> import scipy.special as sc

The coefficient in terms of gamma is equal to \(1/B(a, b)\). Also, when \(x=1\) the integral is equal to \(B(a, b)\). Therefore, \(I_{x=1}(a, b) = 1\) for any \(a, b\).

>>> sc.betainc(0.2, 3.5, 1.0)
1.0

It satisfies \(I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))\), where \(F\) is the hypergeometric function hyp2f1:

>>> a, b, x = 1.4, 3.1, 0.5
>>> x**a * sc.hyp2f1(a, 1 - b, a + 1, x)/(a * sc.beta(a, b))
0.8148904036225295
>>> sc.betainc(a, b, x)
0.8148904036225296

This functions satisfies the relationship \(I_x(a, b) = 1 - I_{1-x}(b, a)\):

>>> sc.betainc(2.2, 3.1, 0.4)
0.49339638807619446
>>> 1 - sc.betainc(3.1, 2.2, 1 - 0.4)
0.49339638807619446