`scipy.stats._multivariate:multinomial_gen`

source: /scipy/stats/_multivariate.py :3881

Signature

class multinomial_gen ( seed = None )

Members

Summary

A multinomial random variable.

Parameters

%(_doc_default_callparams)s
%(_doc_random_state)s

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.

Notes

%(_doc_callparams_note)s

The probability mass function for multinomial is

f (x) = \frac{n !}{x _{1} ! \dots x _{k} !} p_{1}^{x_{1}} \dots p_{k}^{x_{k}},

supported on $x = (x_{1}, \dots, x_{k})$ where each $x_{i}$ is a nonnegative integer and their sum is $n$ .

Examples

from scipy.stats import multinomial
rv = multinomial(8, [0.3, 0.2, 0.5])

✓

rv.pmf([1, 3, 4])

✗

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])
binom.pmf(3, 7, 0.4)

✗

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])

✗

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])
multinomial.pmf([3, 5], 8, p=[.3, .7])

✗

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]])

✗

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

rv = multinomial(n=7, p=[.3, .7])

✓

Aliases

scipy.stats._multivariate.multinomial_gen