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Module « scipy.special »

Fonction betainc - module scipy.special

Signature de la fonction betainc

def betainc(*args, **kwargs) 

Description

help(scipy.special.betainc)

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

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

Regularized incomplete beta function.

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

.. math::

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

for :math:`0 \leq x \leq 1`.

This function is the cumulative distribution function for the beta
distribution; its range is [0, 1].

Parameters
----------
a, b : array_like
       Positive, real-valued parameters
x : array_like
    Real-valued such that :math:`0 \leq x \leq 1`,
    the upper limit of integration
out : ndarray, 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 `beta` from
`scipy.special` to get this "nonregularized" incomplete beta
function by multiplying the result of ``betainc(a, b, x)`` by
``beta(a, b)``.

This function wraps the ``ibeta`` routine 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 :math:`B(a, b)` be the `beta` function.

>>> import scipy.special as sc

The coefficient in terms of `gamma` is equal to
:math:`1/B(a, b)`. Also, when :math:`x=1`
the integral is equal to :math:`B(a, b)`.
Therefore, :math:`I_{x=1}(a, b) = 1` for any :math:`a, b`.

>>> sc.betainc(0.2, 3.5, 1.0)
1.0

It satisfies
:math:`I_x(a, b) = x^a F(a, 1-b, a+1, x)/ (aB(a, b))`,
where :math:`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
:math:`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

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