6 semester






3, Mechanics and Mathematical Modeling


Semester of training







Shtin Sergey Lvovich


Objectives of studying the discipline

To create a knowledge base and skills for students in the field of probability theory

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



Algebra and number theory,

Discrete Math,

Analytic geometry,

Mathematical analysis,

Differential equations,

The theory of functions of a complex variable,

Functional Analysis


Contents of the discipline


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.


Topic 2.1. Conditional probability. Definition of conditional probability. Multiplication theorems. The formula of total probability and the Bayes formula.

Topic 2.2. Independence of events.

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.


Topic 3.1. Random variables and their distributions. Topic 3.2. Classification of random variables. Lebesgue’s theorem. Distribution: binomial, geometric, Poisson, uniform, normal, exponential, chi-square, Student, Fisher, Cauchy, etc. Function and density of distribution.

Topic 3.3. Multidimensional random variables.

Topic 3.4. Independence of random variables. Criteria of Independence.

Theme 3.5. Functional transformations of random variables.


Topic 4.1. Mathematical expectation and its properties. Theme 4.2. Moments of random variables. Dispersion and its properties. Moments of higher orders.

Theme 4.3. Inequalities. Correlation coefficient..

Theme 4.4. Conditional mathematical expectations.


Topic 5.1. Definition and simplest properties. Examples of characteristic functions.

Theme 5.2. Inversion formulas for characteristic functions.

Theme 5.3. Continuity of correspondence between sets of distribution functions and characteristic functions.

Section 6. Limit Theorems.

Topic 6.1. Central limit theorem.

Topic 6.2. Convergence of random variables. 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.



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.

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


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