scipy.special.betaincc#

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

Complement of the regularized incomplete beta function.

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

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

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

betainc: regularized incomplete beta function
betaincinv: inverse of the regularized incomplete beta function
betainccinv: inverse of the complement of the regularized incomplete beta function
beta: beta function
scipy.stats.beta: beta distribution

Notes

Added in version 1.11.0.

Like betainc, betaincc(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. See the betainc docstring for more info on how this impacts limiting cases.

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

betaincc 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

>>> from scipy.special import betaincc, betainc

The naive calculation 1 - betainc(a, b, x) loses precision when the values of betainc(a, b, x) are close to 1:

>>> 1 - betainc(0.5, 8, [0.9, 0.99, 0.999])
array([2.0574632e-09, 0.0000000e+00, 0.0000000e+00])

By using betaincc, we get the correct values:

>>> betaincc(0.5, 8, [0.9, 0.99, 0.999])
array([2.05746321e-09, 1.97259354e-17, 1.96467954e-25])