Math and science::INF ML AI

# Multivariate Gaussian distribution

This card derives the general multivariate normal distribution from the 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:

[$f_X(x) = \quad ?$ ]

#### 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. Then 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, $$f_Z : \mathbb{R}^K \to \mathbb{R}$$:

[\begin{alignat*}{3}f_X(x) &= \frac{1}{\text{rescaling factor} } \; &&f_z(\text{z in terms of x}) \\&= \frac{1}{\text{what?} } &&f_z(\Sigma^{-1}(x - \mu)) \\ &= \frac{1}{\text{what?} } &&(\frac{1}{\sqrt{2\pi} })^K e^{\text{what?} } \\ \end{alignat*}]