7.7. Random Vectors#
We will continue to use the notation of capital letters to denote a random vector. We will specify the space over which the random vector is generated to clarify the dimensionality.
A real random vector \(X\) takes values in the vector space \(\RR^n\). A complex random vector \(Z\) takes values in the vector space \(\CC^n\). We write
The expected value or mean of a random vector is \(\EE(X)\).
Covariance-matrix of a random vector:
We will use the symbols \(\mu\) and \(\Sigma\) for the mean vector and covariance matrix of a random vector \(X\). Clearly
Cross-covariance matrix of two random vectors:
The characteristic function is defined as
The MGF is defined as
The components \(X_1, \dots, X_n\) of a random vector \(X\) are independent if and only if