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Introduction to Probability and Stochastic Processes with Applications

Book Description

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. Coverpage
  2. Titlepage
  3. Copyright
  4. Dedication
  5. Contents in Brief
  6. Contents
  7. Foreword
  8. Preface
  9. Acknowledgments
  10. Introduction
  11. 1 Basic Concepts
    1. 1.1 Probability Space
    2. 1.2 Laplace Probability Space
    3. 1.3 Conditional Probability and Event Independence
    4. 1.4 Geometric Probability
      1. Exercises
  12. 2 Random Variables and Their Distributions
    1. 2.1 Definitions and Properties
    2. 2.2 Discrete Random Variables
    3. 2.3 Continuous Random Variables
    4. 2.4 Distribution of a Function of a Random Variable
    5. 2.5 Expected Value and Variance of a Random Variable
      1. Exercises
  13. 3 Some Discrete Distributions
    1. 3.1 Discrete Uniform, Binomial and Bernoulli Distributions
    2. 3.2 Hypergeometric and Poisson Distributions
    3. 3.3 Geometric and Negative Binomial Distributions
      1. Exercises
  14. 4 Some Continuous Distributions
    1. 4.1 Uniform Distribution
    2. 4.2 Normal Distribution
    3. 4.3 Family of Gamma Distributions
    4. 4.4 Weibull Distribution
    5. 4.5 Beta Distribution
    6. 4.6 Other Continuous Distributions
      1. Exercises
  15. 5 Random Vectors
    1. 5.1 Joint Distribution of Random Variables
    2. 5.2 Independent Random Variables
    3. 5.3 Distribution of Functions of a Random Vector
    4. 5.4 Covariance and Correlation Coefficient
    5. 5.5 Expected Value of a Random Vector and Variance-Covariance Matrix
    6. 5.6 Joint Probability Generating, Moment Generating and Characteristic Functions
      1. Exercises
  16. 6 Conditional Expectation
    1. 6.1 Conditional Distribution
    2. 6.2 Conditional Expectation Given a σ-Algebra
      1. Exercises
  17. 7 Multivariate Normal Distributions
    1. 7.1 Multivariate Normal Distribution
    2. 7.2 Distribution of Quadratic Forms of Multivariate Normal Vectors
      1. Exercises
  18. 8 Limit Theorems
    1. 8.1 The Weak Law of Large Numbers
    2. 8.2 Convergence of Sequences of Random Variables
    3. 8.3 The Strong Law of Large Numbers
    4. 8.4 Central Limit Theorem
      1. Exercises
  19. 9 Introduction to Stochastic Processes
    1. 9.1 Definitions and Properties
    2. 9.2 Discrete-Time Markov Chain
      1. 9.2.1 Classification of States
      2. 9.2.2 Measure of Stationary Probabilities
    3. 9.3 Continuous-Time Markov Chains
    4. 9.4 Poisson Process
    5. 9.5 Renewal Processes
    6. 9.6 Semi-Markov Process
      1. Exercises
  20. 10 Introduction to Queueing Models
    1. 10.1 Introduction
    2. 10.2 Markovian Single-Server Models
      1. 10.2.1 M/M/1/∞ Queueing System
      2. 10.2.2 M/M/1/N Queueing System
    3. 10.3 Markovian MultiServer Models
      1. 10.3.1 M/M/c/∞ Queueing System
      2. 10.3.2 M/M/c/c Loss System
      3. 10.3.3 M/M/c/K Finite-Capacity Queueing System
      4. 10.3.4 M/M/∞ Queueing System
    4. 10.4 Non-Markovian Models
      1. 10.4.1 M/G/1 Queueing System
      2. 10.4.2 GI/M/1 Queueing System
      3. 10.4.3 M/G/1/N Queueing System
      4. 10.4.4 GI/M/1/N Queueing System
      5. Exercises
  21. 11 Stochastic Calculus
    1. 11.1 Martingales
    2. 11.2 Brownian Motion
    3. 11.3 Itô Calculus
      1. Exercises
  22. 12 Introduction to Mathematical Finance
    1. 12.1 Financial Derivatives
    2. 12.2 Discrete-Time Models
      1. 12.2.1 The Binomial Model
      2. 12.2.2 Multi-Period Binomial Model
    3. 12.3 Continuous-Time Models
      1. 12.3.1 Black-Scholes Formula European Call Option
      2. 12.3.2 Properties of Black-Scholes Formula
    4. 12.4 Volatility
      1. Exercises
  23. Appendix A: Basic Concepts on Set Theory
  24. Appendix B: Introduction to Combinatorics
    1. Exercises
  25. Appendix C: Topics on Linear Algebra
  26. Appendix D: Statistical Tables
    1. D.1 Binomial Probabilities
    2. D.2 Poisson Probabilities
    3. D.3 Standard Normal Distribution Function
    4. D.4 Chi-Square Distribution Function
  27. Selected Problem Solutions
  28. References
  29. Glossary
  30. Index