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

This function is the cumulative distribution function for the beta distribution; its range is [0, 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
betaincc: complement of the regularized incomplete beta function
scipy.stats.beta: beta distribution

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).

betainc(a, b, x) is treated as a two parameter family of functions of a single variable x, rather than as a function of three variables. This impacts only the limiting cases a = 0, b = 0, a = inf, b = inf.

In general

\[\lim_{(a, b) \rightarrow (a_0, b_0)} \mathrm{betainc}(a, b, x)\]

is treated as a pointwise limit in x. Thus for example, betainc(0, b, 0) equals 0 for b > 0, although it would be indeterminate when considering the simultaneous limit (a, x) -> (0+, 0+).

This function wraps the ibeta routine from the Boost Math C++ library [2].

betainc has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	✅	✅
Dask	✅	n/a

See Support for the array API standard for more information.

References

[1]

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

[2]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

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