Mathematics for Engineers

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 and queueing 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.

Auteur

Georges Fiche has worked at Alcatel-Lucent for more than 20
years, where he has acted as Technical Coordinator for Performance
Standardization and as Performance Manager for the design of
Alcatel products.

Gerard Hebuterne has been a member of the research lab at
France telecom for 20 years. He is currently a professor and is
responsible for the "Network" Department in INT ( a French school
of higher education).

Résumé
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 and queueing 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.

Contenu

Preface 15

Chapter 1. Probability Theory 19

1.1. Definition and properties of events 19

1.1.1. The concept of an event 19

1.1.2.Complementary events 21

1.1.2.1. Basic properties 21

1.1.3. Properties of operations on events 21

1.1.3.1.Commutativity 21

1.1.3.2.Associativity 21

1.1.3.3.Distributivity 21

1.1.3.4. Difference 21

1.1.3.5.DeMorgan's rules 22

1.2. Probability 23

1.2.1. Definition 23

1.2.2. Basic theorems and results 23

1.2.2.1. Addition theorem 23

1.2.2.2. Conditional probability 24

1.2.2.3. Multiplication theorem 25

1.2.2.4. The posterior probability theorem 26

1.3. Random variable 27

1.3.1. Definition 27

1.3.2. Probability functions of a random variable 27

1.3.2.1.Notations 27

1.3.2.2.Cumulative distribution function 27

1.3.2.3. Probability density function 27

1.3.3. Moments of a random variable 28

1.3.3.1. Moments about the origin 29

1.3.3.2.Central moments 29

1.3.3.3. Mean and variance 29

1.3.3.4.Example applications 31

1.3.4. Couples of random variables 32

1.3.4.1. Definition 32

1.3.4.2. Joint probability 32

1.3.4.3. Marginal probability of couples of random variables 33

1.3.4.4. Conditional probability of a couple of random variables 34

1.3.4.5. Functions of a couple of random variables 34

1.3.4.6. Sum of independent random variables 36

1.3.4.7. Moments of the sum of independent random variables 37

1.3.4.8. Practical interest 39

1.4.Convolution 40

1.4.1. Definition 40

1.4.2. Properties of the convolution operation 41

1.4.2.1.The convolution is commutative 41

1.4.2.2. Convolution of exponential distributions 41

1.4.2.3. Convolution of normal (Gaussian) distributions 41

1.5.Laplace transform 42

1.5.1. Definition 43

1.5.2. Properties 43

1.5.2.1. Fundamental property 43

1.5.2.2. Differentiation property 43

1.5.2.3. Integration property 44

1.5.2.4. Some common transforms 44

1.6. Characteristic function, generating function, z-transform 47

1.6.1.Characteristic function 47

1.6.1.1. Definition 47

1.6.1.2. Inversion formula 48

1.6.1.3. The concept of event indicator and the Heaviside function 48

1.6.1.4. Calculating the inverse function and residues 49

1.6.1.5. The residue theorem 50

1.6.1.6.Asymptotic formula 51

1.6.1.7.Moments 52

1.6.1.8. Some common transforms 53

1.6.2. Generating functions, z-transforms 54

1.6.2.1. Definition 54

1.6.2.2.Moments 54

1.6.2.3. Some common transforms 55

1.6.3.Convolution 56

Chapter 2. Probability Laws 57

2.1.Uniform(discrete) distribution 58

2.2.The binomial law 58

2.3.Multinomial distribution 60

2.4.Geometric distribution 61

2.5. Hypergeometric distribution 62

2.6.The Poisson law 63

2.7. Continuous uniform distribution 65

2.8.Normal (Gaussian) distribution 66

2.9.Chi-2 distribution 70

2.10. Student distribution 71

2.11. Lognormal distribution 72

2.12. Exponential and related distributions 72

2.12.1. Exponential distribution 72

2.12.2. Erlang-k distribution 73

2.12.3. Hyperexponential distribution 75

2.12.4. Generalizing: Coxian distribution 77

2.12.5.Gamma distribution 77

2.12.6.Weibull distribution 78

2.13.Logistic distribution 79

2.14. Pareto distribution 81

2.15.A summary of the main results 82

2.15.1.Discrete distributions 82

2.15.2. Continuous distributions 83 **Chapter 3. St...