Probability Theory

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