These functions provide the density and random number generation for the multivariate normal distribution.

```
dmvn(x, mu, Sigma, log=FALSE)
rmvn(n=1, mu, Sigma)
```

x

This is data or parameters in the form of a vector of length \(k\) or a matrix with \(k\) columns.

n

This is the number of random draws.

mu

This is mean vector \(\mu\) with length \(k\) or matrix with \(k\) columns.

Sigma

This is the \(k \times k\) covariance matrix \(\Sigma\).

log

Logical. If `log=TRUE`

, then the logarithm of the
density is returned.

`dmvn`

gives the density and
`rmvn`

generates random deviates.

Application: Continuous Multivariate

Density: \(p(\theta) = \frac{1}{(2\pi)^{k/2}|\Sigma|^{1/2}} \exp(-\frac{1}{2}(\theta - \mu)'\Sigma^{-1}(\theta - \mu))\)

Inventors: Robert Adrain (1808), Pierre-Simon Laplace (1812), and Francis Galton (1885)

Notation 1: \(\theta \sim \mathcal{MVN}(\mu, \Sigma)\)

Notation 2: \(\theta \sim \mathcal{N}_k(\mu, \Sigma)\)

Notation 3: \(p(\theta) = \mathcal{MVN}(\theta | \mu, \Sigma)\)

Notation 4: \(p(\theta) = \mathcal{N}_k(\theta | \mu, \Sigma)\)

Parameter 1: location vector \(\mu\)

Parameter 2: positive-definite \(k \times k\) covariance matrix \(\Sigma\)

Mean: \(E(\theta) = \mu\)

Variance: \(var(\theta) = \Sigma\)

Mode: \(mode(\theta) = \mu\)

The multivariate normal distribution, or multivariate Gaussian distribution, is a multidimensional extension of the one-dimensional or univariate normal (or Gaussian) distribution. A random vector is considered to be multivariate normally distributed if every linear combination of its components has a univariate normal distribution. This distribution has a mean parameter vector \(\mu\) of length \(k\) and a \(k \times k\) covariance matrix \(\Sigma\), which must be positive-definite.

The conjugate prior of the mean vector is another multivariate normal
distribution. The conjugate prior of the covariance matrix is the
inverse Wishart distribution (see `dinvwishart`

).

When applicable, the alternative Cholesky parameterization should be
preferred. For more information, see `dmvnc`

.

For models where the dependent variable, Y, is specified to be
distributed multivariate normal given the model, the Mardia test (see
`plot.demonoid.ppc`

, `plot.laplace.ppc`

, or
`plot.pmc.ppc`

) may be used to test the residuals.

`dinvwishart`

,
`dmatrixnorm`

,
`dmvnc`

,
`dmvnp`

,
`dnorm`

,
`dnormp`

,
`dnormv`

,
`plot.demonoid.ppc`

,
`plot.laplace.ppc`

, and
`plot.pmc.ppc`

.

```
# NOT RUN {
library(LaplacesDemon)
x <- dmvn(c(1,2,3), c(0,1,2), diag(3))
X <- rmvn(1000, c(0,1,2), diag(3))
joint.density.plot(X[,1], X[,2], color=TRUE)
# }
```

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