SYLLABUS
for doctoral exam
professional field 4.5 Mathematics
(Probability and Mathematical Statistics)
1. Probability and probability spaces. Sigma- additive functions. Carathéodory's extension theorem for
probability measures.
2. Random variables. Distribution functions and probability densities. Multivariate probability
distributions. Integration of random variables.
3. Types of convergence of sequences of random variables. Sequences of independent random variables.
Kolmogorov’s 0-1 law.
4. Laws of large numbers.
5. Central limit theorem (Lindeberg-Feller’s condition).
6. Conditional mathematical expectation and conditional distributions. Markov Chains. Ergodic Theorem.
7. Characteristic functions. Continuity theorem. Theorem of Bochner-Hinchin.
8. Summation of random variables and infinitely-divisible distributions.
9. Renewal theory. Blackwell theorem and the Smith Theorem.
10. Martingales. Markov moments. Doob’s.Theorem
11. Poisson process. Basic properties.
12. Galton-Watson branching processes with one type of particles. Bellman-Harris branching processes.
Asymptotic behavior and limit theorems.
13. Weiner process. Basic properties.
14. Point estimations. Measures of the quality of point estimations. Sufficient statistics. Estimations with
minimal dispersion. Rao-Cramer Inequality. Method of Maximal likelihood.
15. Ordered statistics. Glivenko-Kanteli’s theorem. Kolmogorov-Smirnov Criterion. Nonparametric
criteria.
16. Hypotheses testing. Lemma of Neyman-Pearson. Normal models - t - and F - criteria.
Bibliography
1. Patrick Billingsley, Probability and measure, Third edition, John Wiley and Sons, 1995.
2. William Feller, An introduction to probability theory and its applications, volume II,
Second edition, John Wiley and sons, 1971.
3. William Feller, An introduction to probability theory and its applications, volume I, Third
edition, John Wiley and sons, 1968.
4. David Freedman, Brownian motion and diffusion, Holden-Day, 1971.
5. Geoffrey Grimmett and David Stirzaker, Probability and random processes, 3rd ed., Oxford
University Press, 2001.
6. Samuel Karlin and Howard M. Taylor, A first course in stochastic processes, 2nd ed., Academic
Press, 1975.
7. Joseph Doob, Stochastic processes, Wiley, 1953.
8. K. Athreya. P. Ney. Branching processes. Berlin. Springer-Verlag. 1972.
9. Sheldon M. Ross. Introduction to probability and statistics for engineers and scientists, Third
edition, Elsevier, ISBN: 0-12-598057-4.
The Department of Probability,
Operations Research and Statistics,
Faculty of Mathematics and Informatics
Sofia University “St. Kl. Ohridski”
Sofia, 05.03.2015