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

3

Discrete Random Variables

3.1   Definitions

A random variable assigns numbers to outcomes in the sample space of an experiment.

Chapter 1 defines a probability model. It begins with a physical model of an experiment. An experiment consists of a procedure and observations. The set of all possible observations, S, is the sample space of the experiment. S is the beginning of the mathematical probability model. In addition to S, the mathematical model includes a rule for assigning numbers between 0 and 1 to sets A in S. Thus for every A ⊂ S, the model gives us a probability P[A], where 0 ≤ P[A] ≤ 1.

In this chapter and for most of the remainder of this book, we examine probability models that assign numbers to the outcomes in the sample space. When we observe one of these numbers, we refer to the observation as a random variable. In our notation, the name of a random variable is always a capital letter, for example, X. The set of possible values of X is the range of X. Since we often consider more than one random variable at a time, we denote the range of a random variable by the letter S with a subscript that is the name of the random variable. Thus S_X is the range of random variable X, S_Y is the range of random variable Y, and so forth. We use S_X to denote the range of X because the set of all possible values of X is analogous to S, the set of all possible outcomes of an experiment.

A probability model always begins with an experiment. Each random variable is related directly ...