Introduction

This chapter provides theoretical foundations and examples of of random variables, vectors, and processes. All three concepts are variations on a single theme and may be included in the general term of random object. We will deal specifically with random variables first because they are the simplest conceptually – they can be considered to be special cases of the other two concepts.

Random variables

The name random variable suggests a variable that takes on values randomly. In a loose, intuitive way this is the right interpretation – e.g., an observer who is measuring the amount of noise on a communication link sees a random variable in this sense. We require, however, a more precise mathematical definition for analytical purposes. Mathematically a random variable is neither random nor a variable – it is just a function mapping one sample space into another space. The first space is the sample space portion of a probability space, and the second space is a subset of the real line (some authors would call this a “real-valued” random variable). The careful mathematical definition will place a constraint on the function to ensure that the theory makes sense, but for the moment we informally define a random variable as a function.

A random variable is perhaps best thought of as a measurement on a probability space; that is, for each sample point ω the random variable produces some value, denoted functionally as f(ω).