Table of Contents

Foreword xiii 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 sigma-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 Ito 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

About the Author

LILIANA BLANCO CASTANEDA, 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.

Reviews

"A great strength of this book is the enormous number of detailed examples and the exercises at the end of each chapter, many of which include solutions. The writing style is very clear, because the authors brought their experiences in teaching for several years to its writing...In summary, the first eight chapters provide an excellent introduction to and quick overview of probability theory, with many examples." (Interfaces, 1 September 2013) "The choice of material and the presentation make this book an excellent first introduction into probability theory and stochastic processes from upper undergraduate level onwards in all the areas mentioned above. It may also serve math students at the very initial stages of their studies as a stepping stone to get a sound grasp of some basic concepts of probability." (Contemporary Physics, 13 August 2012)