Consider the i.i.d. Gaussian variables $z_i \sim N(\mu,\sigma)$, where $\mu=0$ and $\sigma=1$. The random variables from a linear combination of $z_i$, e.g.

are correlated. Their covariance is given by

as the covariance between two independent standard normal variables is zero.

Consider a $k$-vector $\vec{z}$ of independent standard normal variables $z_i$, and define the $n$-vector of random variables $\vec{x}$ as the linear combination of the $k$ variables in $\vec{z}$ and a $n$-vector of constants $\vec{\mu}$:

then the covariance matrix of $\vec{x}$ is $\mathbf{\Sigma} = \mathbf{AA}^T$ and the mean is $\vec{\mu}$.

The p.d.f. of a univariate Gaussian variable is

while the p.d.f. of a multivariate Gaussian variable ($n$-vector) is