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.