E-book

# Probability and Random Processes, 2nd Edition

ISBN: 978-1-119-01190-3
528 pages
July 2015

## Description

The second edition enhanced with new chapters, figures, and appendices to cover the new developments in applied mathematical functions

This book examines the topics of applied mathematical functions to problems that engineers and researchers solve daily in the course of their work. The text covers set theory, combinatorics, random variables, discrete and continuous probability, distribution functions, convergence of random variables, computer generation of random variates, random processes and stationarity concepts with associated autocovariance and cross covariance functions, estimation theory and Wiener and Kalman filtering ending with two applications of probabilistic methods. Probability tables with nine decimal place accuracy and graphical Fourier transform tables are included for quick reference. The author facilitates understanding of probability concepts for both students and practitioners by presenting over 450 carefully detailed figures and illustrations, and over 350 examples with every step explained clearly and some with multiple solutions.

Additional features of the second edition of Probability and Random Processes are:

• Updated chapters with new sections on Newton-Pepys’ problem; Pearson, Spearman, and Kendal correlation coefficients; adaptive estimation techniques; birth and death processes; and renewal processes with generalizations
• A new chapter on Probability Modeling in Teletraffic Engineering written by Kavitha Chandra
• An eighth appendix examining the computation of the roots of discrete probability-generating functions

With new material on theory and applications of probability, Probability and Random Processes, Second Edition is a thorough and comprehensive reference for commonly occurring problems in probabilistic methods and their applications.

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Preface for the Second Edition xii

Preface for the First Edition xiv

1 Sets, Fields, and Events 1

1.1 Set Definitions 1

1.2 Set Operations 2

1.3 Set Algebras, Fields, and Events 5

2 Probability Space and Axioms 7

2.1 Probability Space 7

2.2 Conditional Probability 9

2.3 Independence 11

2.4 Total Probability and Bayes’ Theorem 12

3 Basic Combinatorics 16

3.1 Basic Counting Principles 16

3.2 Permutations 16

3.3 Combinations 18

4 Discrete Distributions 23

4.1 Bernoulli Trials 23

4.2 Binomial Distribution 23

4.3 Multinomial Distribution 26

4.4 Geometric Distribution 26

4.5 Negative Binomial Distribution 27

4.6 Hypergeometric Distribution 28

4.7 Poisson Distribution 30

4.8 Newton–Pepys Problem and its Extensions 33

4.9 Logarithmic Distribution 40

4.9.1 Finite Law (Benford’s Law) 40

4.9.2 Infinite Law 43

4.10 Summary of Discrete Distributions 44

5 Random Variables 45

5.1 Definition of Random Variables 45

5.2 Determination of Distribution and Density Functions 46

5.3 Properties of Distribution and Density Functions 50

5.4 Distribution Functions from Density Functions 51

6 Continuous Random Variables and Basic Distributions 54

6.1 Introduction 54

6.2 Uniform Distribution 54

6.3 Exponential Distribution 55

6.4 Normal or Gaussian Distribution 57

7 Other Continuous Distributions 63

7.1 Introduction 63

7.2 Triangular Distribution 63

7.3 Laplace Distribution 63

7.4 Erlang Distribution 64

7.5 Gamma Distribution 65

7.6 Weibull Distribution 66

7.7 Chi-Square Distribution 67

7.8 Chi and Other Allied Distributions 68

7.9 Student-t Density 71

7.10 Snedecor F Distribution 72

7.11 Lognormal Distribution 72

7.12 Beta Distribution 73

7.13 Cauchy Distribution 74

7.14 Pareto Distribution 75

7.15 Gibbs Distribution 75

7.16 Mixed Distributions 75

7.17 Summary of Distributions of Continuous Random Variables 76

8 Conditional Densities and Distributions 78

8.1 Conditional Distribution and Density for P{A} 0 78

8.2 Conditional Distribution and Density for P{A} = 0 80

8.3 Total Probability and Bayes’ Theorem for Densities 83

9 Joint Densities and Distributions 85

9.1 Joint Discrete Distribution Functions 85

9.2 Joint Continuous Distribution Functions 86

9.3 Bivariate Gaussian Distributions 90

10 Moments and Conditional Moments 91

10.1 Expectations 91

10.2 Variance 92

10.3 Means and Variances of Some Distributions 93

10.4 Higher-Order Moments 94

10.5 Correlation and Partial Correlation Coefficients 95

10.5.1 Correlation Coefficients 95

10.5.2 Partial Correlation Coefficients 106

11 Characteristic Functions and Generating Functions 108

11.1 Characteristic Functions 108

11.2 Examples of Characteristic Functions 109

11.3 Generating Functions 111

11.4 Examples of Generating Functions 112

11.5 Moment Generating Functions 113

11.6 Cumulant Generating Functions 115

11.7 Table of Means and Variances 116

12 Functions of a Single Random Variable 118

12.1 Random Variable g(X) 118

12.2 Distribution of Y = g(X) 119

12.3 Direct Determination of Density fY(y) from fX(x) 129

12.4 Inverse Problem: Finding g(X) given fX(x) and fY(y) 132

12.5 Moments of a Function of a Random Variable 133

13 Functions of Multiple Random Variables 135

13.1 Function of Two Random Variables, Z = g(X,Y) 135

13.2 Two Functions of Two Random Variables, Z = g(X,Y), W= h(X,Y) 143

13.3 Direct Determination of Joint Density fZW(z,w) from fXY(x,y) 146

13.4 Solving Z = g(X,Y) Using an Auxiliary Random Variable 150

13.5 Multiple Functions of Random Variables 153

14 Inequalities, Convergences, and Limit Theorems 155

14.1 Degenerate Random Variables 155

14.2 Chebyshev and Allied Inequalities 155

14.3 Markov Inequality 158

14.4 Chernoff Bound 159

14.5 Cauchy–Schwartz Inequality 160

14.6 Jensen’s Inequality 162

14.7 Convergence Concepts 163

14.8 Limit Theorems 165

15 Computer Methods for Generating Random Variates 169

15.1 Uniform-Distribution Random Variates 169

15.2 Histograms 170

15.3 Inverse Transformation Techniques 172

15.4 Convolution Techniques 178

15.5 Acceptance–Rejection Techniques 178

16 Elements of Matrix Algebra 181

16.1 Basic Theory of Matrices 181

16.2 Eigenvalues and Eigenvectors of Matrices 186

16.3 Vector and Matrix Differentiation 190

16.4 Block Matrices 194

17 Random Vectors and Mean-Square Estimation 196

17.1 Distributions and Densities 196

17.2 Moments of Random Vectors 200

17.3 Vector Gaussian Random Variables 204

17.4 Diagonalization of Covariance Matrices 207

17.5 Simultaneous Diagonalization of Covariance Matrices 209

17.6 Linear Estimation of Vector Variables 210

18 Estimation Theory 212

18.1 Criteria of Estimators 212

18.2 Estimation of Random Variables 213

18.3 Estimation of Parameters (Point Estimation) 218

18.4 Interval Estimation (Confidence Intervals) 225

18.5 Hypothesis Testing (Binary) 231

18.6 Bayesian Estimation 238

19 Random Processes 250

19.1 Basic Definitions 250

19.2 Stationary Random Processes 258

19.3 Ergodic Processes 269

19.4 Estimation of Parameters of Random Processes 273

19.4.1 Continuous-Time Processes 273

19.4.2 Discrete-Time Processes 280

19.5 Power Spectral Density 287

19.5.1 Continuous Time 287

19.5.2 Discrete Time 294

20 Classification of Random Processes 320

20.1 Specifications of Random Processes 320

20.1.1 Discrete-State Discrete-Time (DSDT) Process 320

20.1.2 Discrete-State Continuous-Time (DSCT) Process 320

20.1.3 Continuous-State Discrete-Time (CSDT) Process 320

20.1.4 Continuous-State Continuous-Time (CSCT) Process 320

20.2 Poisson Process 321

20.3 Binomial Process 329

20.4 Independent Increment Process 330

20.5 Random-Walk Process 333

20.6 Gaussian Process 338

20.7 Wiener Process (Brownian Motion) 340

20.8 Markov Process 342

20.9 Markov Chains 347

20.10 Birth and Death Processes 357

20.11 Renewal Processes and Generalizations 366

20.12 Martingale Process 370

20.13 Periodic Random Process 374

20.14 Aperiodic Random Process (Karhunen–Loeve Expansion) 377

21 Random Processes and Linear Systems 383

21.1 Review of Linear Systems 383

21.2 Random Processes through Linear Systems 385

21.3 Linear Filters 393

21.4 Bandpass Stationary Random Processes 401

22 Wiener and Kalman Filters 413

22.1 Review of Orthogonality Principle 413

22.2 Wiener Filtering 414

22.3 Discrete Kalman Filter 425

22.4 Continuous Kalman Filter 433

23 Probability Modeling in Traffic Engineering 437

23.1 Introduction 437

23.2 Teletraffic Models 437

23.3 Blocking Systems 438

23.4 State Probabilities for Systems with Delays 440

23.5 Waiting-Time Distribution for M/M/c/∞ Systems 441

23.6 State Probabilities for M/D/c Systems 443

23.7 Waiting-Time Distribution for M/D/c/∞ System 446

23.8 Comparison of M/M/c and M/D/c 448

References 451

24 Probabilistic Methods in Transmission Tomography 452

24.1 Introduction 452

24.2 Stochastic Model 453

24.3 Stochastic Estimation Algorithm 455

24.4 Prior Distribution P{M} 457

24.5 Computer Simulation 458

24.6 Results and Conclusions 460

24.7 Discussion of Results 462

References 462

APPENDICES

A A Fourier Transform Tables 463

B Cumulative Gaussian Tables 467

C Inverse Cumulative Gaussian Tables 472

D Inverse Chi-Square Tables 474

E Inverse Student-t Tables 481

F Cumulative Poisson Distribution 484

G Cumulative Binomial Distribution 488

H Computation of Roots of D(z) = 0 494

References 495

Index 498

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## Author Information

Venkatarama Krishnan, PhD., is Professor Emeritus in the Department of Electrical and Computer Engineering at the University of Massachusetts at Lowell. He has served as a consultant to the Dynamics Research Corporation, the U.S. Department of Transportation, and Bell Laboratories. Dr. Krishnan’s research includes estimation of steady-state queue distribution, tomographic imaging, aerospace, control, communications, and stochastic systems. Dr. Krishnan is a senior member of the IEEE and listed in Who is Who in America.
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