Concepts of Probability Theory

Concepts of Probability Theory

by Paul E. Pfeiffer

Paperback(2d rev. ed)

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This approach to the basics of probability theory employs the simple conceptual framework of the Kolmogorov model, a method that comprises both the literature of applications and the literature on pure mathematics. The author also presents a substantial introduction to the idea of a random process. Intended for college juniors and seniors majoring in science, engineering, or mathematics, the book assumes a familiarity with basic calculus.
After a brief historical introduction, the text examines a mathematical model for probability, random variables and probability distributions, sums and integrals, mathematical expectation, sequence and sums of random variables, and random processes. Problems with answers conclude each chapter, and six appendixes offer supplementary material. This text provides an excellent background for further study of statistical decision theory, reliability theory, dynamic programming, statistical game theory, coding and information theory, and classical sampling statistics.

Product Details

ISBN-13: 9780486636771
Publisher: Dover Publications
Publication date: 06/13/2012
Series: Dover Books on Mathematics Series
Edition description: 2d rev. ed
Pages: 416
Product dimensions: 5.67(w) x 8.21(h) x 0.82(d)

About the Author

Paul E. Pfeiffer is Professor Emeritus of Computational and Applied Mathematics at Rice University. His research interests coincide with his teaching interests: electronic circuits, control systems, analog computers, switching circuits, coding theory, applied probability, and random processes.

Table of Contents

Chapter 1. Introduction
1-1. Basic Ideas and the Classical Definition
1-2. Motivation for a More General Theory
  Selected References
Chapter 2. A Mathematical Model for Probability
2-1. In Search of a Model
2-2. A Model for Events and Their Occurrence
2-3. A Formal Definition of Probability
2-4. An Auxiliary Model-Probability as Mass
2-5. Conditional Probability
2-6. Independence in Probabililty Theory
2-7. Some Techniques for Handling Events
2-8. Further Results on Independent Events
2-9. Some Comments on Strategy
  Selected References
Chapter 3. Random Variables and Probability Distributions
3-1. Random Variables and Events
3-2. Random Variables and Mass Distributions
3-3. Discrete Random Variables
3-4. Probability Distribution Functions
3-5. Families of Random Variables and Vector-valued Random Variables
3-6. Joint Distribution Functions
3-7. Independent Random Variables
3-8. Functions of Random Variables
3-9. Distributions for Functions of Random Variables
3-10. Almost-sure Relationships
  Selected References
Chapter 4. Sums and Integrals
4-1. Integrals of Riemann and Lebesque
4-2. Integral of a Simple Random Variable
4-3. Some Basic Limit Theorems
4-4. Integrable Random Variables
4-5. The Lebesgue-Stieltjes Integral
4-6. Transformation of Integrals
  Selected References
Chapter 5. Mathematical Expectation
5-1. Definition and Fundamental Formulas
5-2. Some Properties of Mathematical Expectation
5-3. The Mean Value of a Random Variable
5-4. Variance and Standard Deviation
5-5. Random Samples and Random Variables
5-6. Probability and Information
5-7. Moment-generating and Characteristic Functions
  Selected References
Chapter 6. Sequences and Sums of Random Variables
6-1. Law of Large Numbers (Weak Form)
6-2. Bounds on Sums of Independent Random Variables
6-3. Types of Convergence
6-4. The Strong Law of Large Numbers
6-5. The Central Limit Theorem
  Selected References
Chapter 7. Random Processes
7-1. The General Concept of a Random Process
7-2. Constant Markov Chains
7-3. Increments of Processes; The Poisson Process
7-4. Distribution Functions for Random Processes
7-5. Processes Consisting of Step Functions
7-6. Expectations; Correlation and Covariance Functions
7-7. Stationary Random Processes
7-8. Expectations and Time Averages; Typical Functions
7-9. Gaussian Random Processes
  Selected References
  Appendix A. Some Elements of Combinatorial Analysis
  Appendix B. Some Topics in Set Theory
  Appendix C. Measurability of Functions
  Appendix D. Proofs of Some Theorems
  Appendix E. Integrals of Complex-valued Random Variables
  Appendix F. Summary of Properties and Key Theorems

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