6 semester






3, Mathematics (economic activity)


Semester of training







Radyno Evgeniy Mefodievich


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;



Algebra and number theory,

Discrete Math,

Analytic geometry,

Mathematical analysis,

Differential equations,

Functional Analysis


Contents of the discipline

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).

Section 7. Fundamentals of the theory of random processes (in the overview 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. Parametric hypotheses. The Neumann-Pearson lemma. Examples. Theme 8.6. Linear Regression and Least Squares.



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.


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.


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





Conditions (requirements), current control

– test;

– colloquium

  The score on the exam is set taking into account: The score on the exam is set taking into account: 30% – work in laboratory and practical classes, 70% – oral examination.


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