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Module « scipy.stats »

Fonction multivariate_hypergeom - module scipy.stats

Signature de la fonction multivariate_hypergeom

def multivariate_hypergeom(m, n, seed=None) 

Description

help(scipy.stats.multivariate_hypergeom)

A multivariate hypergeometric random variable.

Methods
-------
pmf(x, m, n)
    Probability mass function.
logpmf(x, m, n)
    Log of the probability mass function.
rvs(m, n, size=1, random_state=None)
    Draw random samples from a multivariate hypergeometric
    distribution.
mean(m, n)
    Mean of the multivariate hypergeometric distribution.
var(m, n)
    Variance of the multivariate hypergeometric distribution.
cov(m, n)
    Compute the covariance matrix of the multivariate
    hypergeometric distribution.

Parameters
----------
m : array_like
    The number of each type of object in the population.
    That is, :math:`m[i]` is the number of objects of
    type :math:`i`.
n : array_like
    The number of samples taken from the population.
seed : {None, int, np.random.RandomState, np.random.Generator}, optional
    Used for drawing random variates.
    If `seed` is `None`, the `~np.random.RandomState` singleton is used.
    If `seed` is an int, a new ``RandomState`` instance is used, seeded
    with seed.
    If `seed` is already a ``RandomState`` or ``Generator`` instance,
    then that object is used.
    Default is `None`.

Notes
-----
`m` must be an array of positive integers. If the quantile
:math:`i` contains values out of the range :math:`[0, m_i]`
where :math:`m_i` is the number of objects of type :math:`i`
in the population or if the parameters are inconsistent with one
another (e.g. ``x.sum() != n``), methods return the appropriate
value (e.g. ``0`` for ``pmf``). If `m` or `n` contain negative
values, the result will contain ``nan`` there.

The probability mass function for `multivariate_hypergeom` is

.. math::

    P(X_1 = x_1, X_2 = x_2, \ldots, X_k = x_k) = \frac{\binom{m_1}{x_1}
    \binom{m_2}{x_2} \cdots \binom{m_k}{x_k}}{\binom{M}{n}}, \\ \quad
    (x_1, x_2, \ldots, x_k) \in \mathbb{N}^k \text{ with }
    \sum_{i=1}^k x_i = n

where :math:`m_i` are the number of objects of type :math:`i`, :math:`M`
is the total number of objects in the population (sum of all the
:math:`m_i`), and :math:`n` is the size of the sample to be taken
from the population.

.. versionadded:: 1.6.0

Examples
--------
To evaluate the probability mass function of the multivariate
hypergeometric distribution, with a dichotomous population of size
:math:`10` and :math:`20`, at a sample of size :math:`12` with
:math:`8` objects of the first type and :math:`4` objects of the
second type, use:

>>> from scipy.stats import multivariate_hypergeom
>>> multivariate_hypergeom.pmf(x=[8, 4], m=[10, 20], n=12)
0.0025207176631464523

The `multivariate_hypergeom` distribution is identical to the
corresponding `hypergeom` distribution (tiny numerical differences
notwithstanding) when only two types (good and bad) of objects
are present in the population as in the example above. Consider
another example for a comparison with the hypergeometric distribution:

>>> from scipy.stats import hypergeom
>>> multivariate_hypergeom.pmf(x=[3, 1], m=[10, 5], n=4)
0.4395604395604395
>>> hypergeom.pmf(k=3, M=15, n=4, N=10)
0.43956043956044005

The functions ``pmf``, ``logpmf``, ``mean``, ``var``, ``cov``, and ``rvs``
support broadcasting, under the convention that the vector parameters
(``x``, ``m``, and ``n``) are interpreted as if each row along the last
axis is a single object. For instance, we can combine the previous two
calls to `multivariate_hypergeom` as

>>> multivariate_hypergeom.pmf(x=[[8, 4], [3, 1]], m=[[10, 20], [10, 5]],
...                            n=[12, 4])
array([0.00252072, 0.43956044])

This broadcasting also works for ``cov``, where the output objects are
square matrices of size ``m.shape[-1]``. For example:

>>> multivariate_hypergeom.cov(m=[[7, 9], [10, 15]], n=[8, 12])
array([[[ 1.05, -1.05],
        [-1.05,  1.05]],
       [[ 1.56, -1.56],
        [-1.56,  1.56]]])

That is, ``result[0]`` is equal to
``multivariate_hypergeom.cov(m=[7, 9], n=8)`` and ``result[1]`` is equal
to ``multivariate_hypergeom.cov(m=[10, 15], n=12)``.

Alternatively, the object may be called (as a function) to fix the `m`
and `n` parameters, returning a "frozen" multivariate hypergeometric
random variable.

>>> rv = multivariate_hypergeom(m=[10, 20], n=12)
>>> rv.pmf(x=[8, 4])
0.0025207176631464523

See Also
--------
scipy.stats.hypergeom : The hypergeometric distribution.
scipy.stats.multinomial : The multinomial distribution.

References
----------
.. [1] The Multivariate Hypergeometric Distribution,
       http://www.randomservices.org/random/urn/MultiHypergeometric.html
.. [2] Thomas J. Sargent and John Stachurski, 2020,
       Multivariate Hypergeometric Distribution
       https://python.quantecon.org/multi_hyper.html

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