1.1 INTRODUCTION

This book is primarily designed for the study of statistical signal detection and parameter estimation. Such concepts require a good knowledge of the fundamental notions on probability, random variables, and stochastic processes. In Chapter 1, we present concepts on probability and random variables. In Chapter 2, we discuss some important distributions that arise in many engineering applications such as radar and communication systems. Probability theory is a prerequisite for Chapters 3 and 4, in which we cover stochastic processes and some applications. Similarly, the fundamentals of stochastic processes will be essential for proper understanding of the subsequent topics, which cover the fundamentals of signal detection and parameter estimation. Some applications of adaptive thresholding radar constant false alarm rate (CFAR) detection will be presented in Chapter 11. In Chapter 12, we consider the concepts of adaptive CFAR detection using multiple sensors and data fusion. This concept of adaptive thresholding CFAR detection will also be introduced in spread spectrum communication systems.

We start this chapter with the set theory, since it provides the most fundamental concepts in the theory of probability. We introduce the concepts of random variables and probability distributions, statistical moments, two- and higher-dimensional random variables, and the transformation of random variables. We derive some basic results, to which we shall refer throughout the book, and establish the notation to be used.

1.2 SETS AND PROBABILITY