CHAPTER 3
Probability Fundamentals
This chapter reviews the fundamental principles of probability theory that are needed as background for the concepts of random process theory developed in later chapters. The material covers random variables, probability distributions, expected values, change of variables, moment-generating functions, and characteristic functions for both single and multiple random variables. More detailed developments of probability theory from an engineering viewpoint are presented in Refs 1–3.
3.1 ONE RANDOM VARIABLE
The underlying concept in probability theory is that of a set, defined as a collection of objects (also called points or elements) about which it is possible to determine whether any particular object is a member of the set. In particular, the possible outcomes of an experiment (or a measurement) represent a set of points called the sample space. These points may be grouped together in various ways, called events, and under suitable conditions probability functions may be assigned to each. These probabilities always lie between zero and one, the probability of an impossible event being zero and the probability of the certain event being one. Sample spaces are either finite or infinite.
Consider a sample space of points representing the possible outcomes of a particular experiment (or measurement). A random variable x(k) is a set function defined for points k from the sample space; that is, a random variable x(k) is a real number between − ∞ and ...
