# Multivariate Normal Cheatsheet

Multivariate normal (MVN) is used everywhere in machine learning, from simple regressions, linear discriminant analysis, Kalman filters to gaussian processes. But very few textbooks summarize the important characteristics in a concise manner. So here they are.

## Definition

if .

- : dimensional vector
- : dimensional mean vector
- : dimensional covariance matrix. It is positive semi-definite: for all .
- : dimensional matrix.
- where

## Equivalent Definition

if all linear combinations of are univariate normal, i.e.

for any .

## PDF and MGF

- PDF:
- MGF:

## Linear Transformations of MVN

If ,

## Within MVN…

Let , where

and

Then,

- Uncorrelation implies independence:
- Marginal:
- Conditional: , where , .

## Joint Distribution

then

## Leave a Comment