scipy.special.betaincinv#

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

Inverse of the regularized incomplete beta function.

Computes \(x\) such that:

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

where \(I_x\) is the normalized incomplete beta function betainc and \(\Gamma\) is the gamma function [1].

Parameters:

a, barray_like: Positive, real-valued parameters
yarray_like: Real-valued input
outndarray, optional: Optional output array for function values

Returns:

scalar or ndarray: Value of the inverse of the regularized incomplete beta function

See also

betainc: regularized incomplete beta function
gamma: gamma function

Notes

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

Array API Standard Support

betaincinv has support for Python Array API Standard compatible backends in addition to NumPy. 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

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. betaincinv does not currently support out for the PyTorch backend.

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

>>> import scipy.special as sc

This function is the inverse of betainc for fixed values of \(a\) and \(b\).

>>> a, b = 1.2, 3.1
>>> y = sc.betainc(a, b, 0.2)
>>> sc.betaincinv(a, b, y)
0.2
>>>
>>> a, b = 7.5, 0.4
>>> x = sc.betaincinv(a, b, 0.5)
>>> sc.betainc(a, b, x)
0.5