List of Multivariate Distributions

Multinomial

Multinomial distribution having probability mass function

\[f(x) = \dfrac{n!}{x_1!\dots x_k!}p_1^{x_1}\dots p_k^{x_k}\]

for \(x=\{x_1,\dots,x_k\}\) where each \(x_i\) is a non-negative integer and \(\sum_i x_i = n\).

The Multinomial class is imported using the following command:

>>> from UQpy.distributions.collection.Multinomial import Multinomial

class Multinomial(n, p)

Parameters:

• n (Optional[int]) – number of trials

• p (Union[list[float], ndarray]) – probability of a trial falling into each category; should sum to 1

parameters: dict

Ordered list of parameter names, useful when parameter values are stored in vectors and must be passed to the update_params() method.

ordered_parameters: list

Parameters of the distribution.

This attribute is not defined for certain Distribution objects such as those of type JointIndependent or JointCopula. The user is advised to use the get_parameters() method to access the parameters.

The following methods are available for Multinomial:

pmf(), log_pmf(), rvs(), moments() .

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Multivariate Normal

Multivariate normal distribution having probability density function

\[f(x) = \dfrac{1}{\sqrt{(2\pi)^k\det\Sigma}}\exp{-\dfrac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}\]

where \(\mu\) is the mean vector, \(\Sigma\) is the covariance matrix, and \(k\) is the dimension of x.

The MultivariateNormal class is imported using the following command:

>>> from UQpy.distributions.collection.MultivariateNormal import MultivariateNormal

class MultivariateNormal(mean, cov=1.0)

Parameters:

• mean (Union[None, ndarray, list]) – mean vector, numpy.ndarray of shape (dimension, )

• cov (Union[None, int, float, ndarray, list[list]]) – covariance, float or numpy.ndarray of shape (dimension, ) or (dimension, dimension). Default is 1.

parameters: dict

Ordered list of parameter names, useful when parameter values are stored in vectors and must be passed to the update_params() method.

ordered_parameters: list

Parameters of the distribution.

This attribute is not defined for certain Distribution objects such as those of type JointIndependent or JointCopula. The user is advised to use the get_parameters() method to access the parameters.

The following methods are available for MultivariateNormal :

pdf(), log_pdf(), rvs(), fit(), moments().