TABLE OF CONTENTS 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.
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 ? the random variable produces some value, denoted functionally as f( ?). One can view ?
