A collection of problems and exercises on probability theory. (Q2703792)

The book contains about 800 problems and exercises on the main parts of the standard educational course of probability theory and the theory of processes. The contents of the book is as follows: Part I. The probability space. \S 1. Operations of events, \S 2. Classical definition of probability, \S 3. Geometrical probability. Part II. Conditional probabilities and independence. \S 4. Conditional probabilities, \S 5. Independent events, \S 6. Bernoulli trials, \S 7. Total probability formula. Bayes formula. Part III. Random variables and their distributions. \S 8. Random variables, \S 9. Distribution functions, \S 10. The join probability distribution. Independence, \S 11. Moments, \S 12. Characteristic functions, \S 13. Generating functions, \S 14. Infinitely divisible distributions. Part IV. Convergence of random variables and distribution functions. \S 15. Almost sure convergence, \S 16. Convergence in probability, \S 17. Weak convergence, \S 18. Convergence of means and convergence in mean (of order \(\alpha\)). Part V. The law of large numbers. \S 19. Independent identically distributed summands, \S 20. Independent differently distributed summands, \S 21. Dependent summands. Part VI. Limit theorems. \S 22. The central limit theorem, \S 23. Numerical exercises for use of the central limit theorem and the Poisson theorem. Part VII. Markov chains. \S 24. Transition probabilities, \S 25. Classification of states. The ergodicity of chains. Part VIII. Conditional expectation. Martingales. \S 26. Conditional expectation, \S 27. Martingales. Part IX. Random processes. \S 28. General properties, \S 29. The Wiener process, \S 30. The Poisson process, \S 31. The linear theory of random sequences, \S 32. Branching processes with discrete time. Applications (examples of distributions and tables of the normal distribution and the Poisson distribution).