Faculty of BSU

1. |
Title |
THEORY OF PROBABILITY AND MATHEMATICAL STATISTICS |

2. |
Course |
3, Computer mathematics and systems analysis |

3. |
Semester of training |
6 |

4. |
Credits |
4 |

5. |
Lecturer |
Yablonski Aleh Leanidavich |

6. |
Objectives of studying the discipline |
To create a knowledge base and skills for students in the field of probability theory and mathematical statistics acquaint students with the basic principles of probability theory and examples of their applications further formation of students’ skills in abstract mathematical thinking and the ability to apply it in specific tasks, enhancing their mathematical culture As a result of the study, the student should be able to: use the basic laws of random phenomena; apply methods of probability theory and mathematical statistics in other sciences; |

7. |
Requirements |
Algebra and number theory, Discrete Math, Analytic geometry, Mathematical analysis, Differential equations, Functional Analysis |

8. |
Contents of the discipline |
Section 1. PROBABILISTIC SPACES Topic 1.1. Introduction. Topic 1.2. Terminology of probability theory. The subject and problems of probability theory. Events, operations on events. Topic 1.3. Axiomatics of Kolmogorov. Probability properties. Topic 1.4. Examples of probability spaces. Classical, finite, discrete probability spaces. Geometric probability space, the Bertrand paradox. Statistical probability and frequency stability. Section 2. INDEPENDENCE. Topic 2.1. Conditional probability. Definition of conditional probability. Multiplication theorems. The formula of full probability and Bayesian formula. Topic 2.2. Independence of events. Determining the independence of two events and independence in the aggregate of several events. Independence of event classes. Topic 2.3. Independent testing. Bernoulli scheme, polynomial scheme. Topic 2.4. Limit theorems in the Bernoulli scheme. Local and integral limit theorems of Moivre-Laplace and Poisson and their applications. Section 3. RANDOM VALUES. Topic 3.1. Random variables and their distributions. Probability distribution as a measure on a Borel sigma algebra associated with a given random variable. Topic 3.2. Classification of random variables. Lebesgue’s theorem. Distributions: binomial, geometric, Poisson, uniform, normal, exponential, chi-square, Student, Fisher, Cauchy, etc. Function and density of distribution. Topic 3.3. Multidimensional random variables. Properties of the multidimensional distribution function. Classification of multidimensional random variables. Topic 3.4. Independence of random variables. Criteria of independence. Theme 3.5. Functional transformations of random variables. Functions of random variables and the corresponding transformations of the function and the distribution density. The convolution formula. Section 4. Numerical characteristics of random variables. 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). |

9. |
References |
Main literature: 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. Zueў M. M., Syachko Str. Ul. Theory is the iMateNastse i i matematichnaya statyshka. Mazyr: Belaetser, 2000. 6. Ivchenko GI, Medvedev Yu. I. Mathematical statistics. М: Высш. shk., 1984. 7. Lazakovich NV, Stashulyonok 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. Additional literature: 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: Высш. shk., 1984. 16. Lazakovich NV, Stashulyonok 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., Stushulenok 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 Stashulyonok, 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 .: N. V. Lazakovich, E. M. Radyno, S. P. Stasulenok, A. G. Yablonskaya, O.L. Yablonsky]; Ed. NV Lazakovich. – Minsk: BSU, 2014.- 175s. References: 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-M: Science, 1985. |

10. |
Teaching Methods |
interactive methods of teaching (working in small groups (team), problem training) are organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Organized individual, steam and group work. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback |

11. |
Language |
Russian |

12. |
Conditions (requirements), current control |
– test; – colloquium The score on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination. |

13. |
Appraisal Form |
exam |