Probability and Stochastic Processes

Roy Yates received the B.S.E. degree in 1983 from Princeton and the S.M. and Ph.D. degrees in 1986 and 1990 from MIT, all in Electrical Engineering. Since 1990, he has been with the Wireless Information Networks Laboratory (WINLAB) and the ECE department at Rutgers University. Presently, he is an Associate Director of WINLAB and a Professor in the ECE Dept. He is a co-author (with David Goodman) of the text Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers published by John Wiley and Sons. He is a co-recipient (with Christopher Rose and Sennur Ulukus) of the 2003 IEEE Marconi Prize Paper Award in Wireless Communications. His research interests include power control, interference suppression and spectrum regulation for wireless systems.

Features of this Text i Preface vii

1 Experiments, Models, and Probabilities 1

Getting Started with Probability 1

1.1 Set Theory 3

1.2 Applying Set Theory to Probability 7

1.3 Probability Axioms 11

1.4 Conditional Probability 15

1.5 Partitions and the Law of Total Probability 18

1.6 Independence 24

1.7 Matlab 27

Problems 29

2 Sequential Experiments 35

2.1 Tree Diagrams 35

2.2 Counting Methods 40

2.3 Independent Trials 49

2.4 Reliability Analysis 52

2.5 Matlab 55

Problems 57

3 Discrete Random Variables 62

3.1 Definitions 62

3.2 Probability Mass Function 65

3.3 Families of Discrete Random Variables 68

3.4 Cumulative Distribution Function (CDF) 77

3.5 Averages and Expected Value 80

3.6 Functions of a Random Variable 86

3.7 Expected Value of a Derived Random Variable 90

3.8 Variance and Standard Deviation 93

3.9 Matlab 99

Problems 106

4 Continuous Random Variables 118

4.1 Continuous Sample Space 118

4.2 The Cumulative Distribution Function 121

4.3 Probability Density Function 123

4.4 Expected Values 128

4.5 Families of Continuous Random Variables 132

4.6 Gaussian Random Variables 138

4.7 Delta Functions, Mixed Random Variables 145

4.8 Matlab 152

Problems 154

5 Multiple Random Variables 162

5.1 Joint Cumulative Distribution Function 163

5.2 Joint Probability Mass Function 166

5.3 Marginal PMF 169

5.4 Joint Probability Density Function 171

5.5 Marginal PDF 177

5.6 Independent Random Variables 178

5.7 Expected Value of a Function of Two Random Variables 181

5.8 Covariance, Correlation and Independence 184

5.9 Bivariate Gaussian Random Variables 191

5.10 Multivariate Probability Models 195

5.11 Matlab 201

Problems 206

6 Probability Models of Derived Random Variables 218

6.1 PMF of a Function of Two Discrete Random Variables 219

6.2 Functions Yielding Continuous Random Variables 220

6.3 Functions Yielding Discrete or Mixed Random Variables 226

6.4 Continuous Functions of Two Continuous Random Variables 229

6.5 PDF of the Sum of Two Random Variables 232

6.6 Matlab 234

Problems 236

7 Conditional Probability Models 242

7.1 Conditioning a Random Variable by an Event 242

7.2 Conditional Expected Value Given an Event 248

7.3 Conditioning Two Random Variables by an Event 252

7.4 Conditioning by a Random Variable 256

7.5 Conditional Expected Value Given a Random Variable 262

7.6 Bivariate Gaussian Random Variables: Conditional PDFs 265

7.7 Matlab 268

Problems 269

8 Random Vectors 277

8.1 Vector Notation 277

8.2 Independent Random Variables and Random Vectors 280

8.3 Functions of Random Vectors 281

8.4 Expected Value Vector and Correlation Matrix 285

8.5 Gaussian Random Vectors 291

8.6 Matlab 298

Problems 300

9 Sums of Random Variables 306

9.1 Expected Values of Sums 306

9.2 Moment Generating Functions 310

9.3 MGF of the Sum of Independent Random Variables 314

9.4 Random Sums of Independent Random Variables 317

9.5 Central Limit Theorem 321

9.6 Matlab 328

Problems 331

10 The Sample Mean 337

10.1 Sample Mean: Expected Value and Variance 337

10.2 Deviation of a Random Variable from the Expected Value 339

10.3 Laws of Large Numbers 343

10.4 Point Estimates of Model Parameters 345

10.5 Confidence Intervals 352

10.6 Matlab 358

Problems 360

11 Hypothesis Testing 366

11.1 Significance Testing 367

11.2 Binary Hypothesis Testing 370

11.3 Multiple Hypothesis Test 384

11.4 Matlab 387

Problems 389

12 Estimation of a Random Variable 399

12.1 Minimum Mean Square Error Estimation 400

12.2 Linear Estimation of X given Y 404

12.3 MAP and ML Estimation 409

12.4 Linear Estimation of Random Variables from Random Vectors 414

12.5 Matlab 421

Problems 423

13 Stochastic Processes 429

13.1 Definitions and Examples 430

13.2 Random Variables from Random Processes 435

13.3 Independent, Identically Distributed Random Sequences 437

13.4 The Poisson Process 439

13.5 Properties of the Poisson Process 443

13.6 The Brownian Motion Process 446

13.7 Expected Value and Correlation 448

13.8 Stationary Processes 452

13.9 Wide Sense Stationary Stochastic Processes 455

13.10 Cross-Correlation 459

13.11 Gaussian Processes 462

13.12 Matlab 464

Problems 468

Appendix A Families of Random Variables 477

A.1 Discrete Random Variables 477

A.2 Continuous Random Variables 479

Appendix B A Few Math Facts 483

References 489

Index 491