Joint distribution in the context of Gaussian process

In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated.

A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) ${\displaystyle F_{X,Y}(x,y)}$ satisfies

In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons (bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.

Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair ${\displaystyle (X,Y)}$ is from the product of the marginal distributions of ${\displaystyle X}$ and ${\displaystyle Y}$. MI is the expected value of the pointwise mutual information (PMI).