Multivariate Gaussian distribution
This card derives the general multivariate normal distribution from the standard multivariate normal distribution.
Standard multivariate Gaussian/normal distribution
Let
As a vector of random variables
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
Standard multivariate normal as a vector of independent random variables. Proof.
Proof that the above probability density function representsThus, the propability distribution for
Elementwise conceptualization
The quantity
form. Proof.
The Statlect formulation simplifies the expression. Copy-pasted here:
Dispelling the mystery of the co-variance matrix
Here is a screenshot of some notes on the perspective of a multivariate Gaussian random variable being a composition of two standard normal variables. It tries to dispell some of the mystery about why the co-variance matrix comes up the way it does.
Example