Seonglae ChoMultivariate 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
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
