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Fonction multinomial - module scipy.stats

Signature de la fonction multinomial

def multinomial(n, p, seed=None) 

Description

multinomial.__doc__

A multinomial random variable.

    Methods
    -------
    ``pmf(x, n, p)``
        Probability mass function.
    ``logpmf(x, n, p)``
        Log of the probability mass function.
    ``rvs(n, p, size=1, random_state=None)``
        Draw random samples from a multinomial distribution.
    ``entropy(n, p)``
        Compute the entropy of the multinomial distribution.
    ``cov(n, p)``
        Compute the covariance matrix of the multinomial distribution.

    Parameters
    ----------
    x : array_like
        Quantiles, with the last axis of `x` denoting the components.
    n : int
        Number of trials
    p : array_like
        Probability of a trial falling into each category; should sum to 1
    random_state : {None, int, `numpy.random.Generator`,
                    `numpy.random.RandomState`}, optional
    
        If `seed` is None (or `np.random`), the `numpy.random.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``Generator`` or ``RandomState`` instance then
        that instance is used.

    Notes
    -----
    `n` should be a positive integer. Each element of `p` should be in the
    interval :math:`[0,1]` and the elements should sum to 1. If they do not sum to
    1, the last element of the `p` array is not used and is replaced with the
    remaining probability left over from the earlier elements.

    Alternatively, the object may be called (as a function) to fix the `n` and
    `p` parameters, returning a "frozen" multinomial random variable:

    The probability mass function for `multinomial` is

    .. math::

        f(x) = \frac{n!}{x_1! \cdots x_k!} p_1^{x_1} \cdots p_k^{x_k},

    supported on :math:`x=(x_1, \ldots, x_k)` where each :math:`x_i` is a
    nonnegative integer and their sum is :math:`n`.

    .. versionadded:: 0.19.0

    Examples
    --------

    >>> from scipy.stats import multinomial
    >>> rv = multinomial(8, [0.3, 0.2, 0.5])
    >>> rv.pmf([1, 3, 4])
    0.042000000000000072

    The multinomial distribution for :math:`k=2` is identical to the
    corresponding binomial distribution (tiny numerical differences
    notwithstanding):

    >>> from scipy.stats import binom
    >>> multinomial.pmf([3, 4], n=7, p=[0.4, 0.6])
    0.29030399999999973
    >>> binom.pmf(3, 7, 0.4)
    0.29030400000000012

    The functions ``pmf``, ``logpmf``, ``entropy``, and ``cov`` support
    broadcasting, under the convention that the vector parameters (``x`` and
    ``p``) are interpreted as if each row along the last axis is a single
    object. For instance:

    >>> multinomial.pmf([[3, 4], [3, 5]], n=[7, 8], p=[.3, .7])
    array([0.2268945,  0.25412184])

    Here, ``x.shape == (2, 2)``, ``n.shape == (2,)``, and ``p.shape == (2,)``,
    but following the rules mentioned above they behave as if the rows
    ``[3, 4]`` and ``[3, 5]`` in ``x`` and ``[.3, .7]`` in ``p`` were a single
    object, and as if we had ``x.shape = (2,)``, ``n.shape = (2,)``, and
    ``p.shape = ()``. To obtain the individual elements without broadcasting,
    we would do this:

    >>> multinomial.pmf([3, 4], n=7, p=[.3, .7])
    0.2268945
    >>> multinomial.pmf([3, 5], 8, p=[.3, .7])
    0.25412184

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

    >>> multinomial.cov([4, 5], [[.3, .7], [.4, .6]])
    array([[[ 0.84, -0.84],
            [-0.84,  0.84]],
           [[ 1.2 , -1.2 ],
            [-1.2 ,  1.2 ]]])

    In this example, ``n.shape == (2,)`` and ``p.shape == (2, 2)``, and
    following the rules above, these broadcast as if ``p.shape == (2,)``.
    Thus the result should also be of shape ``(2,)``, but since each output is
    a :math:`2 \times 2` matrix, the result in fact has shape ``(2, 2, 2)``,
    where ``result[0]`` is equal to ``multinomial.cov(n=4, p=[.3, .7])`` and
    ``result[1]`` is equal to ``multinomial.cov(n=5, p=[.4, .6])``.

    See also
    --------
    scipy.stats.binom : The binomial distribution.
    numpy.random.Generator.multinomial : Sampling from the multinomial distribution.
    scipy.stats.multivariate_hypergeom :
        The multivariate hypergeometric distribution.