%feature("docstring") OT::Multinomial
"Multinomial distribution.

Available constructors:
    Multinomial(*N=1, P=[0.5]*)

Parameters
----------
N : int
    Number of trials.
P : sequence of float, :math:`p_i \geq 0, i = 1, \ldots, d` and :math:`\sum_{i = 1}^d p_i \leq 1`
    Component probabilities.

Notes
-----
This distribution differs from the classical multinomial distribution definition. The classical multinomial distribution is constrained by :math:`\sum_{i = 1}^d p_i=1` and is supported by :math:`\left\{\vect{x} \in \Nset^d\, |\, \sum_{i = 1}^d x_i = N\right\}`. The OpenTURNS distribution is constrained by :math:`q=\sum_{i = 1}^d p_i \ leq 1` and is supported in general by :math:`\cS=\left\{\vect{x} \in \Nset^d\, |\, \sum_{i = 1}^d x_i \leq N\right\}`.

Its probability density function is defined as:

.. math::

    \Prob{\vect{X} = \vect{x}} = \frac{N!}{x_1! \ldots x_d! (N - s)!}
                                 p_1^{x_1} \ldots p_d^{x_d} (1 - q)^{N - s}\mathbf{1}_{\vect{x} \in \cS}

where :math:`s=\sum_{i = 1}^d x_i`.

If :math:`q=1`, then the distribution generates realizations :math:`\vect{x}=(x_1,\dots,x_d)` such that :math:`\sum_{i = 1}^d x_i = N`. In this case, we recover the classical definition of the multinomial distribution.

If :math:`q<1`, then the distribution generates realizations :math:`\vect{x}=(x_1,\dots,x_d)` such that :math:`\sum_{i = 1}^d x_i \leq N`. It allows to recover the binomial distribution as a special case of multinomial distribution when :math:`d=1`. 

Its first moments are:

.. math::
    :nowrap:

    \begin{eqnarray*}
        \Expect{X_i} & = & N p_i, \quad i = 1, \ldots, n \\
        \Var{X_i} & = & N p_i (1 - p_i), \quad i = 1, \ldots, n \\
        \Cov{X_i, X_j} & = & - N p_i p_j, \quad i, j = 1, \ldots, n, i \neq j
    \end{eqnarray*}

See Also
--------
Dirichlet

Examples
--------
Create a distribution:

>>> import openturns as ot
>>> distribution = ot.Multinomial(1, [0.5])

Draw a sample:

>>> sample = distribution.getSample(5)"

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%feature("docstring") OT::Multinomial::getN
"Accessor to the number of experiments parameter :math:`N`.

Returns
-------
N : int
    Number of experiments :math:`N`."

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%feature("docstring") OT::Multinomial::getP
"Accessor to the component probabilities parameter :math:`\vect{p}`.

Returns
-------
P : :class:`~openturns.Point`
    Component probabilities :math:`\vect{p}`."

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%feature("docstring") OT::Multinomial::setN
"Accessor to the number of experiments parameter :math:`N`.

Parameters
----------
N : int, :math:`\sum_{i = 1}^n x_i \leq N`
    Number of experiments :math:`N`."

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%feature("docstring") OT::Multinomial::setP
"Accessor to the component probabilities parameter :math:`\vect{p}`.

Parameters
----------
P : sequence of float, :math:`0 \leq p_i, i = 1, \ldots, n` and :math:`\sum_{i = 1}^n p_i \leq 1`
    Component probabilities (all positive with sum less than unity)."