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 $(Ω, F, P)$ be a probability space. Let $X : Ω \to R^{K}$ 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) = ?

]

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 $(Ω, F, P)$ be a probability space, and let $Z : Ω \to R^{K}$ be a random vector with a multivariate standard normal distribution. Then let $X = μ + Σ Z$ be another random vector. $X$ has a distribution $f_{X} : R^{K} \to R$ which is a transformed version of $Z$ 's distribution, $f_{Z} : R^{K} \to R$ :

[

\begin{aligned} f_{X} (x) & = \frac{1}{rescaling factor} & f_{z} (z in terms of x) \\ = \frac{1}{what?} & f_{z} (Σ^{- 1} (x - μ)) \\ = \frac{1}{what?} & (\frac{1}{\sqrt{2 π}})^{K} e^{what?} \end{aligned}

]

28.11.2019