Overview
When there are multiple normally distributed random variables, it is common to arrange the variables into a vector ({% \vec{x} %}) and then to define the distribution of the vector. That is, each element {% x_i %} of the vector {% \vec{x} %} represents a normal variable, that has a mean {% \mu_i %} and a variance {% \sigma_i^2 %}
Each variable {% x_i %} also has a covariance with every other variable {% x_j %}. Arranging the means {% \mu_i %} into a vector, and the variances and covariances into a matrix
{% \Sigma =
\begin{bmatrix}
\sigma_{1,1} & \sigma_{1,2} & \sigma_{1,3} \\
\sigma_{2,1} & \sigma_{2,2} & \sigma_{2,3} \\
\sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3} \\
\end{bmatrix}
%}
Then the vector normal distribution is defined as
{% \mathcal{N}(\vec{x}, \vec{\mu}, \Sigma) =
\frac{1}{(2 \pi)^{D/2} | \Sigma |^{1/2}}
exp [- \frac{1}{2} (\vec{x} - \vec{\mu})^T \Sigma ^{-1} (\vec{x} - \vec{\mu})]
%}
Example
Given two normally distributed random variables, a random vector {% \vec{X} %} can be defined as
{% X = \begin{bmatrix}
X_1 \\
X_2 \\
\end{bmatrix} %}
The mean is then a vector defined as
{% \mu =\begin{bmatrix}
\mu_1 \\
\mu_2 \\
\end{bmatrix} %}
with variance matrix given by
{% \sigma =
\begin{bmatrix}
\sigma_{11} & \sigma_{12} \\
\sigma_{21} & \sigma_{22} \\
\end{bmatrix}
%}
Then we have
{% X_1 \sim N(\mu_1, \sigma_{11}) %}
{% X_2 \sim N(\mu_2, \sigma_{22}) %}