Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 3rd Edition

8

Random Vectors

In this chapter, we expand on the concepts presented in Chapter 5. While Chapter 5 introduced the CDF and PDF of n random variables X₁, . . . , X_n, this chapter focuses on the random vector X = [X₁ ... X_n]′. A random vector treats a collection of n random variables as a single entity. Thus, vector notation provides a concise representation of relationships that would otherwise be extremely difficult to represent.

The first section of this chapter presents vector notation for a set of random variables and the associated probability functions. The subsequent sections define marginal probability functions of subsets of n random variables, n independent random variables, independent random vectors, and expected values of functions of n random variables. We then introduce the covariance matrix and correlation matrix, two collections of expected values that play an important role in stochastic processes and in estimation of random variables. The final two sections cover Gaussian random vectors and the application of MATLAB, which is especially useful in working with multiple random variables.

8.1   Vector Notation

A random vector with n dimensions is a concise representation of a set of n random variables. There is a corresponding notation for the probability model (CDF, PMF, or PDF) of a random vector.

When an experiment produces two or more random variables, vector and matrix notation provide a concise representation of probability models and their properties. This ...