Gaussian Random Vectors
Ionuţ Florescu, Ciprian A. Tudor · 2013
In this chapter, the authors develop the most important properties of Gaussian random variables and vectors, namely the moment-generating function, the moments, the joint densities, and the conditional probability densities. The chaos theory developed in the 1970s owes a lot of gratitude to Gaussian processes to be able to produce useful formulas and bounds. Most of the machine learning theory uses assumptions about errors behaving like Gaussian processes. One of the most important properties of a Gaussian vector is the fact that its components are independent if and only if they are uncorrelated. One direction is always true for every random variable: If two random variables are independent, then they are uncorrelated. On the other hand, the converse is strongly related to the structure of the Gaussian vector, and it does not hold in general for other random variables.