Preface vii

1 Random Experiments, Models, and Probabilities 1

Getting Started with Probability 1

1.1 Applying Set Theory to Probability 2

1.2 Probability Axioms 7

1.3 Conditional Probability 10

1.4 Partitions and the Law of Total Probability 13

1.5 Bayes’ Theorem 17

1.6 Independence 18

1.7 Matlab 22

Problems 24

2 Sequential Random Experiments 31

2.1 Tree Diagrams 31

2.2 Counting Methods 35

2.3 Independent Trials 43

2.4 Matlab 46

Problems 48

3 Discrete Random Variables 53

3.1 Definitions 53

3.2 Probability Mass Function 56

3.3 Families of Discrete Random Variables 59

3.4 Cumulative Distribution Function (CDF) 65

3.5 Averages and Expected Value 69

3.6 Functions of a Random Variable 74

3.7 Expected Value of a Derived Random Variable 77

3.8 Variance and Standard Deviation 80

3.9 Matlab 86

Problems 93

4 Continuous Random Variables 103

4.1 Continuous Sample Space 103

4.2 The Cumulative Distribution Function 105

4.3 Probability Density Function 108

4.4 Expected Values 113

4.5 Families of Continuous Random Variables 116

4.6 Gaussian Random Variables 122

4.7 Delta Functions, Mixed Random Variables 128

4.8 Matlab 134

Problems 136

5 Multiple Random Variables 145

5.1 Joint Cumulative Distribution Function 146

5.2 Joint Probability Mass Function 149

5.3 Marginal PMF 152

5.4 Joint Probability Density Function 154

5.5 Marginal PDF 159

5.6 Independent Random Variables 161

5.7 Expected Value of a Function of Two Random Variables 164

5.8 Covariance, Correlation and Independence 167

5.9 Bivariate Gaussian Random Variables 174

5.10 Multivariate Probability Models 178

5.11 Matlab 183

Problems 188

6 Probability Models of Derived Random Variables 199

6.1 PMF of a Function of Two Discrete Random Variables 200

6.2 Functions Yielding Continuous Random Variables 201

6.3 Functions Yielding Discrete or Mixed Random Variables 207

6.4 Continuous Functions of Two Continuous Random Variables 211

6.5 PDF of the Sum of Two Random Variables 214

6.6 Matlab 216

Problems 217

7 Conditional Probability Models 225

7.1 Conditioning a Random Variable by an Event 225

7.2 Conditional Expected Value Given an Event 231

7.3 Conditioning Two Random Variables by an Event 233

7.4 Conditioning by a Random Variable 237

7.5 Conditional Expected Value Given a Random Variable 241

7.6 Bivariate Gaussian Random Variables: Conditional PDFs 245

7.7 Matlab 248

Problems 249

8 Random Vectors 257

8.1 Vector Notation 257

8.2 Independent Random Variables and Random Vectors 260

8.3 Functions of Random Vectors 261

8.4 Expected Value Vector and Correlation Matrix 265

8.5 Gaussian Random Vectors 270

8.6 Matlab 277

Problems 279

9 Sums of Random Variables 285

9.1 Expected Values of Sums 285

9.2 Moment Generating Functions 289

9.3 MGF of the Sum of Independent Random Variables 293

9.4 Characteristic Function and Probability Generating Function 297

9.5 Matlab 301

Problems 304

10 Hypothesis Testing 307

10.1 Significance Testing 308

10.2 Binary Hypothesis Testing 311

10.3 Multiple Hypothesis Test 324

10.4 Matlab 327

Problems 329

11 Estimation of a Random Variable 339

11.1 Minimum Mean Square Error Estimation 339

11.2 Linear Estimation of X given Y 344

11.3 MAP and ML Estimation 349

11.4 Linear Estimation of Random Variables from Random Vectors 353

11.5 Matlab 360

Problems 362

12 Some Probabilistic Inequalities and Bounds 369

12.1 Markov Inequality 369

12.2 Chebyshev’s Inequality 373

12.3 Chernoff Bound 374

12.4 Central Limit Theorem 376

12.5 Sample Mean and Variance 380

12.6 Laws of Large Numbers (LLN) 382

Problems 384

13 Stochastic Processes and Markov Chains 391

13.1 Definitions and Examples 391

13.2 Random Variables from Random Processes 397

13.3 Independent, Identically Distributed Random Sequences 399

13.4 The Poisson Process 400

13.5 Properties of the Poisson Process 404

13.6 The Brownian Motion Process 407

13.7 Markov Process 409

13.8 Discrete-Time Markov Chains 410

13.9 Higher Transition Probabilities: Chapman-Kolmogorov Equations 414

13.10 Long-Run Behavior of Markov Chains 419

13.11 Classification of States of Chains 422

13.12 Markov Chains with Countably Infinite States 426

13.13 Ergodic and Reducible Chains 429

13.14 Birth Process and Death Process 433

13.15 Queuing Models - Poisson Queues 435

13.16 Matlab 441

Problems 447

14 Stationary Processes and Random Signal Processing 457

14.1 Expected Value and Correlation 457

14.2 Stationary Processes 460

14.3 Wide Sense Stationary Processes 463

14.4 Cross-Correlation 466

14.5 Gaussian Processes 469

14.6 Linear Filtering of Continuous-Time Stochastic Processes 471

14.7 Linear Filtering of a Random Sequence 475

14.8 Discrete-Time Linear Filtering: Vectors and Matrices 481

14.9 Power Spectral Density of a Continuous-Time Process 485

14.10 Power Spectral Density of a Random Sequence 490

14.11 Cross Power Spectral Density 494

14.12 Frequency Domain Filter Relationships 496

14.13 Matlab 501

Problems 510

Appendix A The Sample Mean 517

A. 1 Sample Mean: Expected Value and Variance 517

A. 2 Deviation of a Random Variable from the Expected Value 519

A. 3 Laws of Large Numbers 523

A. 4 Point Estimates of Model Parameters 525

A. 5 Confidence Intervals 531

A. 6 Matlab 538

Appendix B Families of Random Variables 541

B. 1 Discrete Random Variables 541

B. 2 Continuous Random Variables 543

Appendix C A Few Math Facts 547

References 553

Index 555