Contents of the discipline
Section 4. NUMERICAL CHARACTERISTICS OF RANDOM VALUES.
Topic 4.1. Mathematical expectation and its properties. Mathematical expectation of a random variable as the Lebesgue integral. Expression for the mathematical expectation of a Borel function of a random variable in terms of the Lebesgue-Stieltjes integral. The property of multiplicativity of mathematical expectations.
Theme 4.2. Moments of random variables. Dispersion and its properties. Moments of higher orders.
Theme 4.3. Inequalities. Correlation coefficient. Correlation coefficient and its properties. Inequalities of Cauchy – Bunyakovskii, Chebyshev, Lyapunov, Jensen.
Theme 4.4. Conditional mathematical expectations. The notion of conditional mathematical expectation (in the overview order).
Section 5. CHARACTERISTIC FUNCTIONS.
Topic 5.1. Definition and simplest properties. Examples of characteristic functions.
Theme 5.2. Inversion formulas for characteristic functions. The uniqueness of the correspondence between the characteristic functions and the corresponding probability distributions.
Theme 5.3. Continuity of correspondence between sets of distribution functions and characteristic functions. Helly’s theorems, direct and inverse limit theorems.
Section 6. Limit Theorems.
Topic 6.1. Central limit theorem. Limit theorem for independent identically distributed summands. The Lindeberg condition. Lyapunov’s theorem.
Topic 6.2. Convergence of random variables. Different types of convergence of random variables (convergence almost sure, convergence in probability convergence in the mean, weak convergence) and the connection between them.
Topic 6.3. Laws of large numbers. The concept of limiting laws, other than normal (in the review order).
Section 7. BASICS OF THE THEORY OF RANDOM PROCESSES (in reviewing order).
Topic 7.1. Definition of a random process. Processes with discrete and continuous time. The trajectories of a random process.
Topic 7.2. Random processes with independent increments. Examples: a Poisson random process and a random process of Brownian motion.
Section 8. ELEMENTS OF MATHEMATICAL STATISTICS.
Theme 8.1. The subject and tasks of mathematical statistics.
Topic 8.2. Basic concepts of sampling theory: sampling, variational series, histogram, frequency range, empirical distribution function, Glivenko’s theorem. Asymptotic normality of sample moments.
Topic 8.3. Estimation of unknown parameters. Consistency (strong consistency) of evaluations. Displaced and unbiased estimates, optimal estimates. Inequality Rao-Kramer. Efficiency. Methods of maximum likelihood and moments. Sufficient statistics. Trust assessment.
Theme 8.4. Testing of statistical hypotheses. Evenly the most powerful criteria.
Topic 8.5. Parametric hypotheses. The Neumann-Pearson lemma. Examples.
Theme 8.6. Linear Regression and Least Squares.
1. Borovkov AA Probability Theory. M .: Nauka, 1986.
2. Wentzel A.D. Course of the theory of random processes. – Moscow: Nauka, 1978.
3. Gikhman II, Skorokhod AV, Yadrenko MI Theory of Probability and Mathematical Statistics. Kiev: Vishcha Shk., 1979.
4. Gnedenko B. V. Course of the theory of probability. Moscow: Nauka, 1988.
5 Zuev M., Sechko Vl. Probability Theory i mathematical statistics. Mozyr: White Wind 2000.6. Ivchenko GI, Medvedev Yu. I. Mathematical statistics. М.: Vysch. shk., 1984.
7. Lazakovich NV, Stashulenok SP, Yablonsky O.L. Theory of Probability: A Textbook. – 3rd ed., With modified. – Minsk: BSU, 2013.
8. Rozanov Yu. A. The theory of probability, random processes, mathematical statistics. Moscow: Nauka, 1985.
9. Sevastyanov BA Course of the theory of probability and mathematical statistics. Moscow: Nauka, 1982.
10. Chistyakov VP Course of the theory of probability. M .: Nauka, 1987.
11. Shiryaev AN Probability. M .: Nauka, 1989.
12. Bolshev LN, Smirnov NV Tables of mathematical statistics. Moscow: Nauka, 1983.
13. AN Kolmogorov, Basic concepts of the theory of probability. Moscow: Nauka, 1974.
14. Kramer G. Mathematical methods of statistics. Moscow: The World, 1976.
15. VM Kruglov. Additional chapters of the theory of probability. М: Vysch. shk., 1984.
16. Lazakovich NV, Stashulenok SP, Yablonsky O.L. Course of probability theory: an electronic textbook. – Minsk: E-book of the Belarusian State University, 2003.
17. Leman E. Testing of statistical hypotheses. Moscow: Nauka, 1964.
18. Partasarati K. Introduction to probability theory and measure theory. Moscow: Mir, 1983.
19. Feller V. Introduction to the theory of probability and its applications. М .: Мир, 1984.Т.1,2.
20. PA Henneken and A. Tortra, Theory of Probability and some of its applications. Moscow: Nauka, 1974.
Collected problems on the discipline “Theory of Probability and Mathematical Statistics”:
21. Zhdanovich VF, Lazakovich NV Radyno N.Ya. Tasks for laboratory work on the course of probability theory and mathematical statistics in two parts. Part 1. Minsk, 1998.
22. Zhdanovich VF, Lazakovich N.V. Radyno N.Ya., Stashulenok S.P. Tasks for laboratory work on the course of probability theory and mathematical statistics in two parts. Part 2. Minsk, 1999.
23. Meshalkin L.D. A collection of problems in probability theory. M: MSU, 1963.
24. Prokhorov AV, Ushakov VG, Ushakov NG Problems on the theory of probability: Basic concepts. Limit theorems. Random processes. M: Science, 1986.
25. Sevastyanov BA, Chistyakov VP, Zubkov AM Collection of problems in probability theory. M: Science, 1989.
26. Theory of Probability: a Workshop: Textbook. allowance for university students. specialist. : at 2 pm Part 1 / [aut .: NV Lazakovich, EM Radyno, SP Stashulenok, SL Shtin, O.L. Yablonsky]; Ed. NV Lazakovich. – Minsk: BSU, 2011. – 147 p.
27. Theory of Probability: a Workshop: Textbook. allowance for university students. specialist. : at 2 pm Part 2 / [aut .: NV Lazakovich, EM Radyno, SP Stashulenok, A. G. Yablonskaya, O.L. Yablonsky]; Ed. NV Lazakovich. – Minsk: BSU, 2014.- 175s.
28. Prokhorov Yu.V., Rozanov Yu.A. Probability theory. М .: Science, 1973
29. A handbook on probability theory and mathematical statistics. Korolyuk VS, Portenko NI, Skorokhod AV, Turbin AF. Moscow: Nauka, 1985.