# Multivariate Normal Distribution¶

The multivariate Normal distribution has the probability density function (PDF):

$f(x\mid \mu ,\Sigma) = (2\pi)^{-\frac{k}{2}} |\Sigma|^{-\frac{1}{2}} \exp \left(-\frac{1}{2}(x-\mu)^\top \Sigma^{-1}(x-\mu) \right)$

where $$\mu \in \mathbb{R}^k$$ is the mean vector and $$\Sigma \in \mathbb{R}^{k\times k}$$ is a positive definite covariance matrix.

## Usage¶

e["Distribution"][*index*]["Type"] = "Multivariate/Normal"

## Configuration¶

These are settings required by this module.

Mean Vector
• Usage: e[“Mean Vector”] = List of real number

• Description: Means of the variables.

Sigma
• Usage: e[“Sigma”] = List of real number

• Description: Cholesky Decomposition of the covariance matrix.

Name
• Usage: e[“Name”] = string

• Description: Defines the name of the distribution.

Random Seed
• Usage: e[“Random Seed”] = unsigned integer

• Description: Defines the random seed of the distribution.

Range
• Usage: e[“Range”] = gsl_rng

• Description: Stores the current state of the distribution in hexadecimal notation.

## Default Configuration¶

These following configuration will be assigned by default. Any settings defined by the user will override the given settings specified in these defaults.

{
"Mean Vector": [],
"Name": "",
"Random Seed": 0,
"Range": "",
"Sigma": []
}