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 Itô 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.

Table of Contents

1. Basic Concepts 1
2. Random Variables and their Distributions 49
3. Some Discrete Distributions 111
4. Some Continuous Distributions 141
5. Random Vectors 189
6. Conditional Expectation 261
7. Multivariate Normal Distribution 291
8. Limit Theorems 307
9. Introduction to Stochastic Processes 333
10. Introduction to Queueing Models 409
11. Stochastic Calculus 453
12. Introduction to Mathematical Finance 489

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