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 Proc |
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 |