scipy.stats.multinomial#

scipy.stats.multinomial = <scipy.stats._multivariate.multinomial_gen object>[source]#

A multinomial random variable.

Parameters:

nint: Number of trials
parray_like: Probability of a trial falling into each category; should sum to 1
seed{None, int, np.random.RandomState, np.random.Generator}, optional: Used for drawing random variates. If seed is None, the 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.

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.

Notes

n should be a positive integer. Each element of p should be in the interval \([0,1]\) 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.

The probability mass function for multinomial is

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

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

New in version 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 \(k=2\) 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 \(2 \times 2\) 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])
>>> # Frozen object with the same methods but holding the given
>>> # degrees of freedom and scale fixed.

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.