Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

3.1 INTRODUCTION

In Chapter 2, we introduced measure theory and defined the probability of event E in the σ-field as a measure that assigns a number from [0, 1] to that event. It is mathematically convenient to map abstract events in to the real line . In most applications, events can be examined directly in without having to consider an underlying abstract sample space Ω. Besides the fact that most problems in engineering are defined and evaluated using numbers (as opposed to abstract quantities in Ω), another reason for defining a random variable on is the importance of intervals for continuous experiments, as discussed in Chapter 2. It is not possible to assign nonzero probabilities to individual points in a continuous experiment; instead, events with nonzero probabilities are defined by intervals on such that the probability assignment satisfies three axioms. The Borel σ-field is ...