Магистратура
2023/2024
Теория вероятностей и математическая статистика
Лучший по критерию «Полезность курса для Вашей будущей карьеры»
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус:
Курс обязательный (Стохастическое моделирование в экономике и финансах)
Направление:
38.04.01. Экономика
Кто читает:
Департамент статистики и анализа данных
Где читается:
Факультет экономических наук
Когда читается:
1-й курс, 1, 2 модуль
Формат изучения:
без онлайн-курса
Охват аудитории:
для своего кампуса
Прогр. обучения:
Стохастическое моделирование в экономике и финансах
Язык:
английский
Кредиты:
6
Контактные часы:
96
Course Syllabus
Abstract
This course aims to provide a solid introduction to probability theory and mathematical statistics. The fundamental concepts and mathematical tools for modeling and analysis of random phenomena will be presented and discussed.
Learning Objectives
- To provide students with the knowledge of the theoretical aspects and modeling tools related to probability theory.
- To provide students with the knowledge of elementary techniques to analyze probabilistic models.
- To present and study some fundamental distributions of random variables.
- To develop students' ability to apply the knowledge acquired during the course and to use probabilistic models in concrete situations, recognizing the appropriate frameworks and analytical tools related to the study.
- To outline the basic concepts and methods of mathematical and applied statistics.
- To give practical skills in applying statistical methods in applied research.
- To give an idea of the applied methods of multidimensional statistical analysis.
Expected Learning Outcomes
- The concepts of random variables will be generalized to random vectors. It will be introduced the concepts of multivariate distributions.
- The students are expected to learn the concept of random variable, some distribution functions, expected value and variance. Also, they will have the opportunity to relate random problems with random variables.
- The students are expected to learn the operations between sets, the notion of σ-algebra, Kolmogorov's axioms and properties of probability measures.
- The students will have the opportunity to model random experiments where the knowledge of continuous random variables is required. The concepts of expected value and variance for continuous random variables will be generalized. Also, the students will learn some of the main inequalities in Probability Theory.
- The students will learn about some convergences that are used in Probability Theory. LLN and CLT and their applications will be discussed.
Course Contents
- Introduction to Probability Theory
- Discrete random variables
- Continuous random variables
- Random vectors
- Convergences, LLN and CLT
- Statistical estimation of parameters. Samples. Property of estimators. Unbiasedness, efficiency, consistency
- Interval estimation. Standard confidence intervals for the parameters of a normal population. Confidence intervals for the mean, variance, difference of means, variances ratio, population proportion, the difference of proportions. The sample size
- Hypotheses testing. Type I, type II errors. P-value of the test. Tests on the values of the parameters of the normal population. Tests on mean, variance, differences, difference of means, variances ratio, population
- Estimation methods. The Method of Moments. Maximum Likelihood method. Their properties, examples. Information inequality (Fisher)
- Test statistics, Neumann-Pearson Lemma. Likelihood ratio test. Wald test. Lagrange multiplier test
- Goodness-of-fit tests. Contingency tables. Kolmogorov-Smirnov test
- Bayes approach to estimation
- One- and Two-way ANOVA
- Some concepts of non-parametric methods. Wilcoxon tests, run test. Rank correlation coefficients
- Sufficient statistic. Minimal sufficient statistic. Rao-Blackwell Theorem. Complete statistics. Lehmann-Sheffe Theorems
Interim Assessment
- 2023/2024 2nd moduleThe final grade will be an average of the two modules/part of the course, with 50% for the part "Probability Theory" and 50% for the part "Mathematical Statistics". For module 1: 45% * N1 + 55% * N2 where N2=End of module Exam, N1= weighted averaged of homework assignments, seminar participation and mid-module test
Bibliography
Recommended Core Bibliography
- Gut, A. (2005). Probability: A Graduate Course. New York, NY: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=155835
- Introduction to mathematical statistics, Hogg, R. V., 2014
- Ross, S. M. (2010). Introduction to Probability Models (Vol. 10th ed). Amsterdam: Elsevier Ltd. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=334597
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
- Exercises in probability : a guided tour from measure theory to random processes, via conditioning, Chaumont, L., 2012
- Linde, W. (2017). Probability Theory : A First Course in Probability Theory and Statistics. [N.p.]: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1438416
- Stroock, D. W. (2011). Probability Theory : An Analytic View (Vol. 2nd ed). New York: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=357430
- Xia, X.-G. (2019). A Simple Introduction to Free Probability Theory and its Application to Random Matrices. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.1902.10763