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Handbook of Probability

Book Description

THE COMPLETE COLLECTION NECESSARY FOR A CONCRETE UNDERSTANDING OF PROBABILITY

Written in a clear, accessible, and comprehensive manner, the Handbook of Probability presents the fundamentals of probability with an emphasis on the balance of theory, application, and methodology. Utilizing basic examples throughout, the handbook expertly transitions between concepts and practice to allow readers an inclusive introduction to the field of probability.

The book provides a useful format with self-contained chapters, allowing the reader easy and quick reference. Each chapter includes an introduction, historical background, theory and applications, algorithms, and exercises. The Handbook of Probability offers coverage of:

  • Probability Space

  • Probability Measure

  • Random Variables

  • Random Vectors in Rn

  • Characteristic Function

  • Moment Generating Function

  • Gaussian Random Vectors

  • Convergence Types

  • Limit Theorems

The Handbook of Probability is an ideal resource for researchers and practitioners in numerous fields, such as mathematics, statistics, operations research, engineering, medicine, and finance, as well as a useful text for graduate students.

Table of Contents

  1. Cover
  2. Wiley Handbooks in Applied Statistics
  3. Title Page
  4. Copyright
  5. List of Figures
  6. Preface
  7. Introduction
  8. Chapter One: Probability Space
    1. 1.1 Introduction/Purpose of the Chapter
    2. 1.2 Vignette/Historical Notes
    3. 1.3 Notations and Definitions
    4. 1.4 Theory and Applications
    5. 1.5 Summary
    6. Exercises
  9. Chapter Two: Probability Measure
    1. 2.1 Introduction/Purpose of the Chapter
    2. 2.2 Vignette/Historical Notes
    3. 2.3 Theory and Applications
    4. 2.4 Lebesgue Measure on the Unit Interval (0,1]
    5. Exercises
  10. Chapter Three: Random Variables: Generalities
    1. 3.1 Introduction/Purpose of the Chapter
    2. 3.2 Vignette/Historical Notes
    3. 3.3 Theory and Applications
    4. Exercises
  11. Chapter Four: Random Variables: The Discrete Case
    1. 4.1 Introduction/Purpose of the Chapter
    2. 4.2 Vignette/Historical Notes
    3. 4.3 Theory and Applications
    4. 4.4 Examples of Discrete Random Variables
    5. Exercises
  12. Chapter Five: Random Variables: The Continuous Case
    1. 5.1 Introduction/Purpose of the Chapter
    2. 5.2 Vignette/Historical Notes
    3. 5.3 Theory and Applications
    4. 5.4 Examples
    5. Exercises
  13. Chapter Six: Generating Random Variables
    1. 6.1 Introduction/Purpose of the Chapter
    2. 6.2 Vignette/Historical Notes
    3. 6.3 Theory and Applications
    4. 6.4 Generating Multivariate Distributions with Prescribed Covariance Structure
    5. Exercises
  14. Chapter Seven: Random Vectors in Rn
    1. 7.1 Introduction/Purpose of the Chapter
    2. 7.2 Vignette/Historical Notes
    3. 7.3 Theory and Applications
    4. Exercises
  15. Chapter Eight: Characteristic Function
    1. 8.1 Introduction/Purpose of the Chapter
    2. 8.2 Vignette/Historical Notes
    3. 8.3 Theory and Applications
    4. 8.4 Calculation of the Characteristic Function for Commonly Encountered Distributions
    5. Exercises
  16. Chapter Nine: Moment-Generating Function
    1. 9.1 Introduction/Purpose of the Chapter
    2. 9.2 Vignette/Historical Notes
    3. 9.3 Theory and Applications
    4. Exercises
  17. Chapter Ten: Gaussian Random Vectors
    1. 10.1 Introduction/Purpose of the Chapter
    2. 10.2 Vignette/Historical Notes
    3. 10.3 Theory and Applications
    4. Exercises
  18. Chapter Eleven: Convergence Types. Almost Sure Convergence. Lp-Convergence. Convergence in Probability
    1. 11.1 Introduction/Purpose of the Chapter
    2. 11.2 Vignette/Historical Notes
    3. 11.3 Theory and Applications: Types of Convergence
    4. 11.4 Relationships Between Types of Convergence
    5. Exercises
  19. Chapter Twelve: Limit Theorems
    1. 12.1 Introduction/Purpose of the Chapter
    2. 12.2 Vignette/Historical Notes
    3. 12.3 Theory and Applications
    4. 12.4 Central Limit Theorem
    5. Exercises
  20. Chapter Thirteen: Appendix A: Integration Theory. General Expectations
    1. 13.1 Integral of Measurable Functions
    2. 13.2 General Expectations and Moments of a Random Variable
  21. Chapter Fourteen: Appendix B: Inequalities Involving Random Variables and Their Expectations
    1. 14.1 Functions of Random Variables. The Transport Formula
  22. Bibliography
  23. Index