Signal Detection and Estimation, Second Edition
Packed with over 2,100 equations, 230 figures and 183 problems, this authoritative resource helps you apply the material to projects in the field involving signal processing, radar, and communications.
TABLE OF CONTENTS Signal Detection and Estimation, Second Edition
Chapter 2: Distributions
2.1 INTRODUCTION
In the previous chapter, we have defined the concepts of probability, random variables, and statistical moments. In this chapter, we shall study some important distribution functions that are frequently encountered. Since these distributions have a wide range of applications, we shall study them in their general form, and in some cases, we give more details for particular applications. Some of the notions defined will be applied to these special distributions, which yield some standard results to be used later. In Sections 2.2 and 2.3, we present some discrete and continuous distribution functions, respectively. Special distribution functions will be presented in Section 2.4.
2.2 DISCRETE RANDOM VARIABLES
2.2.1 The Bernoulli, Binomial, and Multinomial Distributions
The simplest distribution is one with only two possible events. For example, a coin is tossed, and the events are heads or tails, which must occur with some probability. Tossing the coin n times consists of a series of independent trials, each of which yields one of the two possible outcomes: heads or tails. These two possible outcomes are also referred to as "success" associated with the value 1 and "failure" associated with the value 0. Since all experiments are assumed to be identical, the outcome 1 occurs with probability p, whereas the outcome 0 occurs with probability 1 ? p, with 0 ? p ? 1. These are called the Bernoulli trials.
A random variable X is said to have a Bernoulli distribution if for some p
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