dist_multivariate_normal: The multivariate normal distribution

In distributional: Vectorised Probability Distributions

The multivariate normal distribution

Description

The multivariate normal distribution is a generalization of the univariate normal distribution to higher dimensions. It is widely used in multivariate statistics and describes the joint distribution of multiple correlated continuous random variables.

Usage

dist_multivariate_normal(mu = 0, sigma = list(diag(1)))

Arguments

mu	A list of numeric vectors for the distribution's mean.
sigma	A list of matrices for the distribution's variance-covariance matrix.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multivariate_normal.html

In the following, let \mathbf{X} be a k-dimensional multivariate normal random variable with mean vector mu = \boldsymbol{\mu} and variance-covariance matrix sigma = \boldsymbol{\Sigma}.

Support: \mathbf{x} \in \mathbb{R}^k

Mean: \boldsymbol{\mu}

Variance-covariance matrix: \boldsymbol{\Sigma}

Probability density function (p.d.f):

f(\mathbf{x}) = \frac{1}{(2\pi)^{k/2} |\boldsymbol{\Sigma}|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1}(\mathbf{x} - \boldsymbol{\mu})\right)

where |\boldsymbol{\Sigma}| is the determinant of \boldsymbol{\Sigma}.

Cumulative distribution function (c.d.f):

P(\mathbf{X} \le \mathbf{q}) = P(X_1 \le q_1, \ldots, X_k \le q_k)

The c.d.f. does not have a closed-form expression and is computed numerically.

Moment generating function (m.g.f):

M(\mathbf{t}) = E(e^{\mathbf{t}^T \mathbf{X}}) = \exp\left(\mathbf{t}^T \boldsymbol{\mu} + \frac{1}{2}\mathbf{t}^T \boldsymbol{\Sigma} \mathbf{t}\right)

See Also

mvtnorm::dmvnorm(), mvtnorm::pmvnorm(), mvtnorm::qmvnorm(), mvtnorm::rmvnorm()

Examples

dist <- dist_multivariate_normal(mu = list(c(1,2)), sigma = list(matrix(c(4,2,2,3), ncol=2)))
dimnames(dist) <- c("x", "y")
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, cbind(2, 1))
density(dist, cbind(2, 1), log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7, kind = "equicoordinate")
quantile(dist, 0.7, kind = "marginal")

distributional documentation built on June 23, 2026, 5:08 p.m.