Магистратура
2026/2027
Теория вероятностей
Статус:
Курс обязательный (Аналитика данных и прикладная статистика / Data Analytics and Social Statistics)
Где читается:
Факультет социальных наук
Когда читается:
1-й курс, 2 модуль
Охват аудитории:
для своего кампуса
Преподаватели:
Валиулин Валерий Эрижанович
Язык:
английский
Кредиты:
3
Контактные часы:
40
Course Syllabus
Abstract
The objective of the discipline "Probability Theory" is to lay the foundation of probability concepts and methodology for all courses that follow in the Data Analytics and Social Statistics programme. The course is strongly related and complementary to other compulsory courses provided in the first year (e.g. Introduction to Statistics, Applied Linear Models, Contemporary Data Analysis) and sets a crucial prerequisite for later courses and research projects as well as for the master thesis.
Learning Objectives
- To provide students with a rigorous theoretical foundation in probability theory, including the axiomatic framework of probability, discrete and continuous random variables, their distributions, moment-generating functions, and the fundamental limit theorems (the Law of Large Numbers and the Central Limit Theorem) that govern the behaviour of sums of random variables.
- To develop students' ability to translate real-world problems involving randomness into well-defined probabilistic models, select the appropriate distributional framework, and apply the correct inferential tools based on the structure of the problem, the nature of dependence, and the available information.
- To equip students with hands-on skills in computing and interpreting probabilities, expectations, variances, and covariances for multivariate settings, as well as in using statistical software to simulate and analyse probabilistic models, thereby laying the groundwork for applied data analytics, subsequent quantitative courses, and independent research.
Expected Learning Outcomes
- Be able to define and apply the concepts of sample space, events, probability, random variables, and their distributions.
- Be able to formulate and apply the definitions of convergence in distribution and in probability, formulate scientific problems involving randomness in mathematical terms.
- Be able to formulate and apply theorems concerning functions of random variables and the moment- generating functions, Chebyshev’s theorem, the Central Limit Theorem and the Law of Large Numbers.
- Be able to use probability in future courses and analytical career overall.
- Have an understanding of the basic principles of probability and lay the foundation for future learning in the area.
- Have the skill to meaningfully develop an appropriate model for the research question, using probability theory.
- Have the skill to work with statistical software, required to analyze the data.
- Know joint probability distributions, expectation, variance and covariance of random variables.
- Know the basic principles of using probability for using analytic models.
- Know the role of probability theory in the sciences, communicate the ideas and results of probability.
Course Contents
- Introduction. Probabilty Models and Axioms
- Bayes rule. Independence
- Discrete random variables I
- Discrete random variables II
- Discrete random variables. Problems
- Continous random variables
- Multiple Continuous Random Variables
- Functions of multiple random variables. Correlation
- Weak law of large numbers
- Central limit theorem
Interim Assessment
- 2026/2027 2nd module0.25 * Homework Assignment 1 + 0.25 * Homework Assignment 2 + 0.5 * Final exam
Bibliography
Recommended Core Bibliography
- A first look at rigorous probability, Rosenthal, J. S., 2020
- Chaumont, L., & Yor, M. (2012). Exercises in Probability : A Guided Tour From Measure Theory to Random Processes, Via Conditioning (Vol. 2nd ed). Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=466664
- Introductory Probability - CCBY4_056 - Charles Grinstead & J. Laurie Snell - 2022 - Open Educational Resources: libretexts.org - https://ibooks.ru/products/390842 - 390842 - iBOOKS
Recommended Additional Bibliography
- A. Ya. Dorogovtsev, D. S. Silvestrov, A. V. Skorokhod, & M. I. Yadrenko. (2018). Probability Theory: Collection of Problems. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1790324
- Biswas, D. (2019). Probability and Statistics: Volume I. [N.p.]: New Central Book Agency. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2239779
- Huber, F. (2019). A Logical Introduction to Probability and Induction. New York, NY: Oxford University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1944071
- Probability and random processes, Grimmett, G. R., 2020
- Probability and statistical inference, Hogg, R. V., 2020