Introduction to Probability and Stochastic Processes with Applications

An easily accessible, real-world approach to probability and stochastic processes Introduction to Probability and Stochastic Processes with Applications presents a clear, easy-to-understand treatment of probability and stochastic processes, providing readers with a solid foundation they can build upon throughout their careers. With an emphasis on applications in engineering, applied sciences, business and finance, statistics, mathematics, and operations research, the book features numerous real-world examples that illustrate how random phenomena occur in nature and how to use probabilistic techniques to accurately model these phenomena. The authors discuss a broad range of topics, from the basic concepts of probability to advanced topics for further study, including It integrals, martingales, and sigma algebras. Additional topical coverage includes: Distributions of discrete and continuous random variables frequently used in applications Random vectors, conditional probability, expectation, and multivariate normal distributions The laws of large numbers, limit theorems, and convergence of sequences of random variables Stochastic processes and related applications, particularly in queueing systems Financial mathematics, including pricing methods such as risk-neutral valuation and the Black-Scholes formula Extensive appendices containing a review of the requisite mathematics and tables of standard distributions for use in applications are provided, and plentiful exercises, problems, and solutions are found throughout. Also, a related website features additional exercises with solutions and supplementary material for classroom use. Introduction to Probability and Stochastic Processes with Applications is an ideal book for probability courses at the upper-undergraduate level. The book is also a valuable reference for researchers and practitioners in the fields of engineering, operations research, and computer science who conduct data analysis to make decisions in their everyday work.

An easily accessible, real-world approach to probability andstochastic processesIntroduction to Probability and Stochastic Processes withApplications presents a clear, easy-to-understand treatment ofprobability and stochastic processes, providing readers with asolid foundation they can build upon throughout their careers. Withan emphasis on applications in engineering, applied sciences,business and finance, statistics, mathematics, and operationsresearch, the book features numerous real-world examples thatillustrate how random phenomena occur in nature and how to useprobabilistic techniques to accurately model these phenomena.The authors discuss a broad range of topics, from the basicconcepts of probability to advanced topics for further study,including Itô integrals, martingales, and sigma algebras.Additional topical coverage includes:* Distributions of discrete and continuous random variablesfrequently used in applications* Random vectors, conditional probability, expectation, andmultivariate normal distributions* The laws of large numbers, limit theorems, and convergence ofsequences of random variables* Stochastic processes and related applications, particularly inqueueing systems* Financial mathematics, including pricing methods such asrisk-neutral valuation and the Black-Scholes formulaExtensive appendices containing a review of the requisitemathematics and tables of standard distributions for use inapplications are provided, and plentiful exercises, problems, andsolutions are found throughout. Also, a related website featuresadditional exercises with solutions and supplementary material forclassroom use. Introduction to Probability and StochasticProcesses with Applications is an ideal book for probabilitycourses at the upper-undergraduate level. The book is also avaluable reference for researchers and practitioners in the fieldsof engineering, operations research, and computer science whoconduct data analysis to make decisions in their everyday work.

Foreword xiiiPreface xvAcknowledgments xviiIntroduction xix1. Basic Concepts 11.1 Probability Space 11.2 Laplace Probability Space 131.3 Conditional Probability and Event Independence 181.4 Geometric Probability 34Exercises 362. Random Variables and their Distributions 492.1 Definitions and Properties 492.2 Discrete Random Variables 592.3 Continuous Random Variables 642.4 Distribution of a Function of a Random Variable 692.5 Expected Value and Variance of a Random Variable 77Exercises 973. Some Discrete Distributions 1113.1 Discrete Uniform, Binomial and Bernoulli Distributions1113.2 Hypergeometric and Poisson Distributions 1193.3 Geometric and Negative Binomial Distributions 128Exercises 1334. Some Continuous Distributions 1414.1 Uniform Distribution 1414.2 Normal Distribution 1474.3 Family of Gamma Distribution 1584.4 Weibull Distribution 1674.5 Beta Distribution 1694.6 Other Continuous Distributions 173Exercises 1785. Random Vectors 1895.1 Joint Distribution of Random Variables 1895.2 Independent Random Variables 2065.3 Distribution of Functions of a Random Vector 2145.4 Covariance and Correlation Coefficient 2245.5 Expected Value and Variance of a Random Vector 2325.6 Generating Functions 236Exercises 2476. Conditional Expectation 2616.1 Conditional Distribution 2616.2 Conditional Expectation given a sigma¯-algebra276Exercises 2837. Multivariate Normal Distribution 2917.1 Multivariate Normal Distribution 2917.2 Distribution of Quadratic Forms of Multivariate NormalVectors 298Exercises 3048. Limit Theorems 3078.1 The Weak Law of Large Numbers 3078.2 Convergence of Sequences of Random Variables 3138.3 The Strong Law of Large Numbers 3168.4 Central Limit Theorem 323Exercises 3289. Introduction to Stochastic Processes 3339.1 Definitions and Properties 3349.2 Discrete Time Markov Chain 3389.3 Continuous Time Markov Chains 3649.4 Poisson Process 3749.5 Renewal Processes 3839.6 Semi-Markov process 393Exercises 39910. Introduction to Queueing Models 40910.1 Introduction 40910.2 Markovian Single Server Models 41110.3 Markovian Multi Server Models 42310.4 Non-Markovian Models 432Exercises 44911. Stochastic Calculus 45311.1 Martingales 45311.2 Brownian Motion 46411.3 Itô Calculus 473Exercises 48412. Introduction to Mathematical Finance 48912.1 Financial Derivatives 49012.2 Discrete-time Models 49612.3 Continuous-time models 51312.4 Volatility 523Exercises 525Appendix A. Basic Concepts on Set Theory 529Appendix B. Introduction to Combinatorics 535Appendix C. Topics on Linear Algebra 545Appendix D. Statistical Tables 547Problem Solutions 559References 575Bibliography 575Glossary 579Index 583