jax.scipy.special.betainc¶

jax.scipy.special.betainc(a, b, x)[source]¶

Incomplete beta function.

LAX-backend implementation of betainc(). Original docstring below.

betainc(x1, x2, x3, /, out=None, *, where=True, casting=’same_kind’, order=’K’, dtype=None, subok=True[, signature, extobj])

betainc(a, b, x, out=None)

Computes the 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

b (a,) – Positive, real-valued parameters
x (array-like) – Real-valued such that \(0 \leq x \leq 1\), the upper limit of integration

Returns

Value of the incomplete beta function

Return type

array-like

See also

beta(): beta function
betaincinv(): inverse of the incomplete beta function

Notes

The incomplete beta function is also sometimes defined without the gamma terms, in which case the above definition is the so-called regularized incomplete beta function. Under this definition, you can get the incomplete beta function by multiplying the result of the SciPy function by beta.

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