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 - betafrom- scipy.specialto 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- 0for- b > 0, although it would be indeterminate when considering the simultaneous limit- (a, x) -> (0+, 0+).- This function wraps the - ibetaroutine from the Boost Math C++ library [2].- betainchas 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=1and 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 - 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