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

5

Multiple Random Variables

Chapter 3 and Chapter 4 analyze experiments in which an outcome is one number. Beginning with this chapter, we analyze experiments in which an outcome is a collection of numbers. Each number is a sample value of a random variable. The probability model for such an experiment contains the properties of the individual random variables and it also contains the relationships among the random variables. Chapter 3 considers only discrete random variables and Chapter 4 considers only continuous random variables. The present chapter considers all random variables because a high proportion of the definitions and theorems apply to both discrete and continuous random variables. However, just as with individual random variables, the details of numerical calculations depend on whether random variables are discrete or continuous. Consequently, we find that many formulas come in pairs. One formula, for discrete random variables, contains sums, and the other formula, for continuous random variables, contains integrals.

In this chapter, we consider experiments that produce a collection of random variables, X₁, X₂, … , X_n, where n can be any integer. For most of this chapter, we study n = 2 random variables: X and Y. A pair of random variables is enough to show the important concepts and useful problem-solving techniques. Moreover, the definitions and theorems we introduce for X and Y generalize to n random variables. These generalized definitions appear near the end ...