Multivariate Random Variable - Expected Value, Covariance, and Cross-covariance

Expected Value, Covariance, and Cross-covariance

The expected value or mean of a random vector X is a fixed vector E(X) whose elements are the expected values of the respective random variables.

The covariance matrix (also called the variance-covariance matrix) of an n× 1 random vector is an n × n matrix whose i, j element is the covariance between the ith and the jth random variables. The covariance matrix is the expected value, element by element, of the n × n matrix computed as T, where the superscript T refers to the transpose of the indicated vector:

By extension, the cross-covariance matrix between two random vectors X and Y (X having n elements and Y having p elements) is the n × p matrix

where again the indicated matrix expectation is taken element-by-element in the matrix. The cross-covariance matrix Cov(Y, X) is simply the transpose of the matrix Cov(X, Y).