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
betabeta function
betaincinvinverse of the regularized incomplete beta function
betaincccomplement of the regularized incomplete beta function
scipy.stats.betabeta 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
betafromscipy.specialto get this “nonregularized” incomplete beta function by multiplying the result ofbetainc(a, b, x)bybeta(a, b).This function wraps the
ibetaroutine from the Boost Math C++ library [2].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
betafunction.>>> import scipy.special as sc
The coefficient in terms of
gammais 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