Introduction to Probability and Stochastic Processes With Applications

LILIANA BLANCO CASTAÑEDA, DrRerNat, is Associate Professor in the Department of Statistics at the National University of Colombia and the author of several journal articles and three books on basic and advanced probability.

VISWANATHAN ARUNACHALAM, PhD, is Associate Professor in the Department of Mathematics at the Universidad de los Andes, Colombia. He has published numerous journal articles in areas such as optimization, stochastic processes, and the mathematics of financial derivatives.

SELVAMUTHU DHARMARAJA, PhD, is Associate Professor in the Department of Mathematics and the Bharti School of Telecommunication Technology and Management at the Indian Institute of Technology Delhi. The author of several journal articles, he is Associate Editor for the International Journal of Communication Systems.

Preface xv

Acknowledgments xvii

Introduction xix

1. Basic Concepts 1

1.1 Probability Space 1

1.2 Laplace Probability Space 13

1.3 Conditional Probability and Event Independence 18

1.4 Geometric Probability 34

Exercises 36

2. Random Variables and their Distributions 49

2.1 Definitions and Properties 49

2.2 Discrete Random Variables 59

2.3 Continuous Random Variables 64

2.4 Distribution of a Function of a Random Variable 69

2.5 Expected Value and Variance of a Random Variable 77

Exercises 97

3. Some Discrete Distributions 111

3.1 Discrete Uniform, Binomial and Bernoulli Distributions 111

3.2 Hypergeometric and Poisson Distributions 119

3.3 Geometric and Negative Binomial Distributions 128

Exercises 133

4. Some Continuous Distributions 141

4.1 Uniform Distribution 141

4.2 Normal Distribution 147

4.3 Family of Gamma Distribution 158

4.4 Weibull Distribution 167

4.5 Beta Distribution 169

4.6 Other Continuous Distributions 173

Exercises 178

5. Random Vectors 189

5.1 Joint Distribution of Random Variables 189

5.2 Independent Random Variables 206

5.3 Distribution of Functions of a Random Vector 214

5.4 Covariance and Correlation Coefficient 224

5.5 Expected Value and Variance of a Random Vector 232

5.6 Generating Functions 236

Exercises 247

6. Conditional Expectation 261

6.1 Conditional Distribution 261

6.2 Conditional Expectation given a σ^-algebra 276

Exercises 283

7. Multivariate Normal Distribution 291

7.1 Multivariate Normal Distribution 291

7.2 Distribution of Quadratic Forms of Multivariate Normal Vectors 298

Exercises 304

8. Limit Theorems 307

8.1 The Weak Law of Large Numbers 307

8.2 Convergence of Sequences of Random Variables 313

8.3 The Strong Law of Large Numbers 316

8.4 Central Limit Theorem 323

Exercises 328

9. Introduction to Stochastic Processes 333

9.1 Definitions and Properties 334

9.2 Discrete Time Markov Chain 338

9.3 Continuous Time Markov Chains 364

9.4 Poisson Process 374

9.5 Renewal Processes 383

9.6 Semi-Markov process 393

Exercises 399

10. Introduction to Queueing Models 409

10.1 Introduction 409

10.2 Markovian Single Server Models 411

10.3 Markovian Multi Server Models 423

10.4 Non-Markovian Models 432

Exercises 449

11. Stochastic Calculus 453

11.1 Martingales 453

11.2 Brownian Motion 464

11.3 Itô Calculus 473

Exercises 484

12. Introduction to Mathematical Finance 489

12.1 Financial Derivatives 490

12.2 Discrete-time Models 496

12.3 Continuous-time models 513

12.4 Volatility 523

Exercises 525

Appendix A. Basic Concepts on Set Theory 529

Appendix B. Introduction to Combinatorics 535

Appendix C. Topics on Linear Algebra 545

Appendix D. Statistical Tables 547

Problem Solutions 559

References 575

Bibliography 575

Glossary 579

Index 583

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