Probability Theory

Autorentext

Y. A. Rozanov

Klappentext

This book, a concise introduction to modern probability theory and certain of its ramifications, deals with a subject indispensable to natural scientists and mathematicians alike. Here the readers, with some knowledge of mathematics, will find an excellent treatment of the elements of probability together with numerous applications. Professor Y. A. Rozanov, an internationally known mathematician whose work in probability theory and stochastic processes has received wide acclaim, combines succinctness of style with a judicious selection of topics. His book is highly readable, fast-moving, and self-contained.
The author begins with basic concepts and moves on to combination of events, dependent events and random variables. He then covers Bernoulli trials and the De Moivre-Laplace theorem, which involve three important probability distributions (binomial, Poisson, and normal or Gaussian). The last three chapters are devoted to limit theorems, a detailed treatment of Markov chains, continuous Markov processes. Also included are appendixes on information theory, game theory, branching processes, and problems of optimal control. Each of the eight chapters and four appendixes has been equipped with numerous relevant problems (150 of them), many with hints and answers.
This volume is another in the popular series of fine translations from the Russian by Richard A. Silverman. Dr. Silverman, a former member of the Courant Institute of Mathematical Sciences of New York University and the Lincoln Laboratory of the Massachusetts Institute of Technology, is himself the author of numerous papers on applied probability theory. He has heavily revised the English edition and added new material. The clear exposition, the ample illustrations and problems, the cross-references, index, and bibliography make this book useful for self-study or the classroom.

Inhalt

1 BASIC CONCEPTS

1. Probability and Relative Frequency
2. Rudiments of Combinatorial Analysis
   Problems
   2 COMBINATION OF EVENTS
3. Elementary Events. The Sample Space
4. The Addition Law for Probabilities
   Problems
   3 DEPENDENT EVENTS
5. Conditional Probability
6. Statistical Independence
   Problems
   4 RANDOM VARIABLES
7. Discrete and Continuous Random Variables. Distribution Functions
8. Mathematical Expectation
9. Chebyshev's Inequality. The Variance and Correlation
   Coefficient
   Problems
   5 THREE IMPORTANT PROBABILITY DISTRIBUTIONS
10. Bernoulli Trials. The Binomial and Poisson Distributions
11. The De Moivre-Laplace Theorem. The Normal Distribution
    Problems
    6 SOME LIMIT THEOREMS
12. The Law of Large Numbers
13. Generating Functions. Weak Convergence of Probability Distributions
14. Characteristic Functions. The Central Limit Theorem
    Problems
    7 MARKOV CHAINS
15. Transition Probabilities
16. Persistent and Transient States
17. Limiting Probabilities. Stationary Distributions
    Problems
    8 CONTINUOUS MARKOV PROCESSES
18. Definitions. The Sojourn Time
19. The Kolmogorov Equations
20. More on Limiting Probabilities. Erlang's Formula
    Problems
    APPENDIX 1 INFORMATION THEORY
    APPENDIX 2 GAME THEORY
    APPENDIX 3 BRANCHING PROCESSES
    APPENDIX 4 PROBLEMS OF OPTIMAL CONTROL
    BIBLIOGRAPHY
    INDEX