# Multivariate Gaussian distribution

The *general* multivariate normal distribution can be conceptualized
as the distribution of a random variable which is a linear transformation of a
another random variable having a standard multivariate normal distribution.

### Standard multivariate Gaussian/normal distribution

Let \( (\Omega, \mathrm{F}, \mathbb{P}) \) be a probability space. Let
\( X : \Omega \to \mathbb{R}^K \) be a continuous random vector. \( X \) is
said to have a *standard multivariate normal distribution*
iff its joint probability density function is:

#### As a vector of random variables

\( X \) can be considered to be a vector of independent random variables, each having a standard normal distribution. The proof of this formulation on the reverse side.

### General multivariate

The general multivariate normal distribution is best understood as being the distribution that results from applying a linear transformation to a random variable having a multivariate standard normal distribution.

### General multivariate normal distribution

Let \( (\Omega, \mathrm{F}, \mathbb{P}) \) be a probability space, and let \( Z : \Omega \to \mathrm{R}^K \) be a random vector with a multivariate standard normal distribution. The let \( X = \mu + \Sigma Z \) be another random vector. \( X \) has a distribution \( f_X : \mathbb{R}^K \to \mathbb{R} \) which is a transformed version of \( Z \)'s distribution, \( f_Z : \mathbb{R}^K \to \mathbb{R} \):