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Probability, Random Variables, and Random Processes

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Beschreibung

Probability, Random Variables, and Random Processes is a comprehensive textbook on probability theory for engineers that provides a more rigorous mathematical framework than is usually encountered in undergraduate courses. It is intended for first-year graduate students who have some familiarity with probability and random variables, though not necessarily of random processes and systems that operate on random signals. It is also appropriate for advanced undergraduate students who have a strong mathematical background.

The book has the following features:

  • Several appendices include related material on integration, important inequalities and identities, frequency-domain transforms, and linear algebra. These topics have been included so that the book is relatively self-contained. One appendix contains an extensive summary of 33 random variables and their properties such as moments, characteristic functions, and entropy.
  • Unlike most books on probability, numerous figures have been included to clarify and expand upon important points. Over 600 illustrations and MATLAB plots have been designed to reinforce the material and illustrate the various characterizations and properties of random quantities.
  • Sufficient statistics are covered in detail, as is their connection to parameter estimation techniques. These include classical Bayesian estimation and several optimality criteria: mean-square error, mean-absolute error, maximum likelihood, method of moments, and least squares.
  • The last four chapters provide an introduction to several topics usually studied in subsequent engineering courses: communication systems and information theory; optimal filtering (Wiener and Kalman); adaptive filtering (FIR and IIR); and antenna beamforming, channel equalization, and direction finding. This material is available electronically at the companion website.

Probability, Random Variables, and Random Processes is the only textbook on probability for engineers that includes relevant background material, provides extensive summaries of key results, and extends various statistical techniques to a range of applications in signal processing.



JOHN J. SHYNK, PhD, is Professor of Electrical and Computer Engineering at the University of California, Santa Barbara. He was a Member of Technical Staff at Bell Laboratories, and received degrees in systems engineering, electrical engineering, and statistics from Boston University and Stanford University.



Autorentext

JOHN J. SHYNK, PhD, is Professor of Electrical and Computer Engineering at the University of California, Santa Barbara. He was a Member of Technical Staff at Bell Laboratories, and received degrees in systems engineering, electrical engineering, and statistics from Boston University and Stanford University.



Klappentext

An engineering perspective on probability and random processes

Probability, Random Variables, and Random Processes is a comprehensive textbook on probability theory for engineers that provides a more rigorous mathematical framework than is usually encountered in undergraduate courses. It is intended for first-year graduate students who have some familiarity with probability and random variables, though not necessarily of random processes and systems that operate on random signals. It is also appropriate for advanced undergraduate students who have a strong mathematical background.

The book has the following features:

  • Several appendices include related material on integration, important inequalities and identities, frequency-domain transforms, and linear algebra. These topics have been included so that the book is relatively self-contained. One appendix contains an extensive summary of 33 random variables and their properties such as moments, characteristic functions, and entropy.
  • Unlike most books on probability, numerous figures have been included to clarify and expand upon important points. Over 600 illustrations and MATLAB plots have been designed to reinforce the material and illustrate the various characterizations and properties of random quantities.
  • Sufficient statistics are covered in detail, as is their connection to parameter estimation techniques. These include classical Bayesian estimation and several optimality criteria: mean-square error, mean-absolute error, maximum likelihood, method of moments, and least squares.
  • The last four chapters provide an introduction to several topics usually studied in subsequent engineering courses: communication systems and information theory; optimal filtering (Wiener and Kalman); adaptive filtering (FIR and IIR); and antenna beamforming, channel equalization, and direction finding. This material is available electronically at the companion website.

Probability, Random Variables, and Random Processes is the only textbook on probability for engineers that includes relevant background material, provides extensive summaries of key results, and extends various statistical techniques to a range of applications in signal processing.



Zusammenfassung

Probability, Random Variables, and Random Processes is a comprehensive textbook on probability theory for engineers that provides a more rigorous mathematical framework than is usually encountered in undergraduate courses. It is intended for first-year graduate students who have some familiarity with probability and random variables, though not necessarily of random processes and systems that operate on random signals. It is also appropriate for advanced undergraduate students who have a strong mathematical background.

The book has the following features:

  • Several appendices include related material on integration, important inequalities and identities, frequency-domain transforms, and linear algebra. These topics have been included so that the book is relatively self-contained. One appendix contains an extensive summary of 33 random variables and their properties such as moments, characteristic functions, and entropy.
  • Unlike most books on probability, numerous figures have been included to clarify and expand upon important points. Over 600 illustrations and MATLAB plots have been designed to reinforce the material and illustrate the various characterizations and properties of random quantities.
  • Sufficient statistics are covered in detail, as is their connection to parameter estimation techniques. These include classical Bayesian estimation and several optimality criteria: mean-square error, mean-absolute error, maximum likelihood, method of moments, and least squares.
  • The last four chapters provide an introduction to several topics usually studied in subsequent engineering courses: communication systems and information theory; optimal filtering (Wiener and Kalman); adaptive filtering (FIR and IIR); and antenna beamforming, channel equalization, and direction finding. This material is available electronically at the companion website.

Probability, Random Variables, and Random Processes is the only textbook on probability for engineers that includes relevant background material, provides extensive summaries of key results, and extends various statistical techniques to a range of applications in signal processing.



Inhalt
PREFACE xxi

NOTATION xxv

1 Overview and Background 1

1.1 Introduction 1

1.1.1 Signals, Signal Processing, and Communications 3

1.1.2 Probability, Random Variables, and Random Vectors 9

1.1.3 Random Sequences and Random Processes 11

1.1.4 Delta Functions 16

1.2 Deterministic Signals and Systems 19

1.2.1 Continuous Time 20

1.2.2 Discrete Time 25

1.2.3 Discrete-Time Filters 29

1.2.4 State-Space Realizations 32

1.3 Statistical Signal Processing with MATLAB® 35

1.3.1 Random Number Generation 35

1.3.2 Filtering 38

Problems 39

Further Reading 45

PART I Probability, Random Variables, and Expectation

2 Probability Theory 49

2.1 Introduction 49

2.2 Sets and Sample Spaces 50

2.3 Set Operations 54

2.4 Events and Fields 58

2.5 Summary of a Random Experiment 64

2.6 Measure Theory 64

2.7 Axioms of Probability 68

2.8 Basic Probability Results 69

2.9 Conditional Probability 71

2.10 Independence 73

2.11 Bayes' Formula 74

2.12 Total Probability 76

2.13 Discrete Sample Spaces 79

2.14 Continuous Sample Spaces 83

2.15 Nonmeasurable Subsets of R 84

Problems 87

Further Reading 90

3 Random Variables 91

3.1 Introduction 91

3.2 Functions and Mappings 91

3.3 Distribution Function 96

3.4 Probability Mass Function 101

3.5 Probability Density Function 103

3.6 Mixed Distributions 104

3.7 Parametric Models for Random Variables 107

3.8 Continuous Random Variables 109

3.8.1 Gaussian Random Variable (Normal) 110

3.8.2 Log-Normal Random Variable 113

3.8.3 Inverse Gaussian Random Variable (Wald) 114

3.8.4 Exponential Random Variable (One-Sided) 116

3.8.5 Laplace Random Variable (Double-Sided Exponential) 119

3.8.6 Cauchy Random Variable 122

3.8.7 Continuous Uniform Random Variable 124

3.8.8 Triangular Random Variable 125

3.8.9 Rayleigh Random Variable 127

3.8.10 Rice Random Variable 129

3.8.11 Gamma Random Variable (Erlang for r ? N) 131

3.8.12 Beta Random Variable (Arcsine for ? = ? = 1/2, Power Function for ? = 1) 133

3.8.13 Pareto Random Variable 136

3.8.14 Weibull Random Variable 137

3.8.15 Logistic Random Variable (Sigmoid for {? = 0, ? = 1}) 139

3.8.16 Chi Random Variable (MaxwellBoltzmann, Half-Normal) 141

3.8.17 Chi-Square Random Variable 144

3.8.18 F-Distribution 147

3.8.19 Student's t Distribution 149

3.8.20 Extreme Value Distribution (Type I: Gumbel) 150

3.9 Discrete Random Variables 151

3.9.1 Bernoulli Random Variable 152

3.9.2 Binomial Random Variable 154

3.9.3 Geometric Random Variable (with Support Z+ or N) 157

3.9.4 Negative Binomial Random Variable (Pascal) 160

3.9.5 Poisson Random Variable 162

3.9.6 Hypergeometric Random Variable 165

3.9.7 Discrete Uniform Random Variable 167

3.9.8 Logarithmic Random Variable (Log-Series) 168

3.9.9 Zeta Random Variable (Zipf) 170

Problems 173

Further Reading 176

4 Multiple Random Variables 177

4.1 Introduction 177

4.2 Random Variable Approximations 177

4.2.1 Binomial Approximation of Hypergeometric 177

4.2.2 Poisson Approximation of Binomial 179

4.2.3 Gaussian Approximations 181

4.2.4 Gaussian Approximation of Binomial 181

4.2.5 Gaussian Approximation of Poisson 181

4.2.6 Gaussian Approximation of Hypergeometric 183

4.3 Joint and Marginal Distributions 183

4.4 Independent Random Variables 186

4.5 Conditional Distribution 187

4.6 Random Vectors 190

4.6.1 Bivariate Uniform Distribution 193

4.6.2 Multivariate Gaussian Distribution 193

4.6.3 Multivariate Student's t Distribution 196

4.6.4 Multinomial Distribution 197

4.6.5 Multivariate Hypergeometric Distribution 198

4.6.6 Bivariate Exponential Distributions 200

4.7 Generating Dependent Random Variables 201

4.8 Random Variable Transformations 205

4.8.1 Transformations of Discrete Random Variables 205

4.8.2 Transformations of Continuous Random Variables 207

4.9 Important Functions of Two Random Variables 218

4.9.1 Sum: Z = X + Y 218

4.9.2 Difference: Z = X ? Y 220

4.9.3 Product: Z = XY 221

4.9.4 Quotient (Ratio): Z = X/Y 224

4.10 Transformations of Random Variable Families 226

4.10.1 Gaussian Transformations 226

4.10.2 Exponential Transformations 227

4.10.3 Chi-Square Transformations 228

4.11 Transformations of Random Vectors 229

4.12 Sample Mean ¯X and Sample Variance S2 232

4.13 Minimum, Maximum, and Order Statistics 234

4.14 Mixtures 2...

Produktinformationen

Titel: Probability, Random Variables, and Random Processes
Untertitel: Theory and Signal Processing Applications
Autor:
EAN: 9781118393949
ISBN: 978-1-118-39394-9
Digitaler Kopierschutz: Adobe-DRM
Format: E-Book (pdf)
Herausgeber: Wiley-Interscience
Genre: Programmiersprachen
Anzahl Seiten: 794
Veröffentlichung: 17.09.2012
Jahr: 2012
Untertitel: Englisch
Dateigrösse: 33.6 MB