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 aRandomState
orGenerator
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 nonnegative 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 \(n\).
Added 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
, andcov
support broadcasting, under the convention that the vector parameters (x
andp
) 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,)
, andp.shape == (2,)
, but following the rules mentioned above they behave as if the rows[3, 4]
and[3, 5]
inx
and[.3, .7]
inp
were a single object, and as if we hadx.shape = (2,)
,n.shape = (2,)
, andp.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 sizep.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,)
andp.shape == (2, 2)
, and following the rules above, these broadcast as ifp.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)
, whereresult[0]
is equal tomultinomial.cov(n=4, p=[.3, .7])
andresult[1]
is equal tomultinomial.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.