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Concepts of Probability Theory


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Table of Contents

Preface 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 Problems 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 Problems 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 Problems 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 Problems 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 Problems Selected References Appendixes 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 BIBLIOGRAPHY INDEX

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.

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