Mathematics for Engineers

Georges Fiche, Gerard Hebuterne

• Publisher: Wiley-ISTE
• Year: 2013
• ISBN:
  9781118623336
• Edition: 1
• Topic: Computer Science
• Subtopic: Programming Algorithms
• Index: Computer Science

This book offers comprehensive coverage of all the mathematical tools needed by engineers in the field of processing and transport of all forms of information, data and images - as well as many other engineering disciplines.It provides essential theories, equations and results in probability theory and statistics, which constitute the basis for the presentation of signal processing, information theory, traffic andqueueing theory, and reliability. The mathematical foundations of simulation are also covered.

The book's accessible style will enable students, engineers and researches new to this area to advance their knowledge of communication and other engineering technologies; however, it will also serve as a useful reference guide to anyone wishing to further explore this field.

Chapter 1

Probability Theory

In this chapter we introduce the fundamental concepts of probability theory. Indeed, telecommunication systems are mainly concerned with basically unpredictable phenomena. In order to analyze a signal, as well as to process information or to characterize the performance and reliability of equipment, the engineer is permanently faced with random phenomena. This explains the need to handle them using the tools and methods of probability theory.

We present the notions and properties of random variables, and the major theorems underpinning the theory. We also develop the properties of the transforms of distribution functions, which appear as essential tools in solving complex probabilistic problems.

1.1. Definition and properties of events

1.1.1. The concept of an event

Probability theory is mainly built on the concept of random events. The particular outcome of an experiment cannot be predicted, although the set of all possible outcomes is known: any particular outcome will be called a “random event”.

The concept of a random event is quite natural. Consider the traditional example consisting of tossing a die: the outcome is that one of the six faces will come up, numbered 1,2, …, 6. “I toss the die and 3 comes up” is an event. Each such occurrence is an elementary event. Six different outcomes are possible corresponding to any one of the six faces of the die coming up. Nothing is known about the next outcome, except that it belongs to a certain set, one which is usually able to be described. The set of all the possible outcomes (here, the six possible faces coming up) is the event space, or the sample space, and is usually denoted as Ω.

The formalism of set theory method is useful when describing events and their properties. Indeed, the properties of events may be stated in terms of the traditional operations of the set theory. This last provides methods to analyze combinations of events in order to predict properties of more complex events. The terminology and correspondence are summarized below in Table 1.1.

Table 1.1. Events and sets

Event	Any subset of the sample space
Sure event	Ω: the sure event is the occurrence of anyone of the elements of Ω
Impossib...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Chapter 1: Probability Theory
• Chapter 2: Probability Laws
• Chapter 3: Statistics
• Chapter 4: Signal Theory
• Chapter 5: Information and Coding Theory
• Chapter 6: Traffic and Queueing Theory
• Chapter 7: Reliability Theory
• Chapter 8: Simulation
• Appendix: Mathematical Refresher
• Bibliography
• Index