Multivariate Gaussian

Multivariate Normal Distribution, Joint Normal Distribution

It is a single distribution and GMM is multiple.

Each random variable normally distributed, at the same time joint multi-variable normally distributed

Covariance Matrix affects exponential part of

Probability Density Function

Marginals

Posterior

where

Conditional Mean

How the mean of shifts when is given

generalized form with minus except notation

generalized form with range notation

multi to multi with dimension analysis if we choose k variables for left side

Generalized version with a set notation

Conditional Variance

generalized form with minus except notation

generalized form with range notation

multi to multi with dimension analysis if we choose k variables for left side

Generalized version with a set notation

Inverse matrix

If the covariance matrix is singular, it means that the random variables are fully constrained or deterministic, not truly random. For example, if , then the distribution of would collapse into a plane, not a proper 3D Gaussian distribution.

2-variables

3-variables

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables, each of which clusters around a mean value.

https://en.wikipedia.org/wiki/Multivariate_normal_distribution