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Bachelor 2022/2023

# Mathematics and Statistics

Type: Elective course (Political Science and World Politics)
Area of studies: Political Science
When: 1 year, 3, 4 module
Mode of studies: offline
Open to: students of one campus
Language: English
ECTS credits: 7
Contact hours: 80

### Course Syllabus

#### Abstract

The goal of the course Mathematics and Statistics is to study basic concepts of algebra, geometry, analysis, probability theory and mathematical statistics at the level that would allow students to solve applied problems that require the use of mathematical apparatus. The course materials can be used for the further study and application of methods aimed at solving problems from different fields of knowledge, for the definition and study of mathematical models for such problems. This discipline will allow the students of the Department of Applied Political Science to study the mathematical and statistical components of their professional education.

#### Learning Objectives

• The goal of this course is to introduce the students to the basic mathematical notions and techniques needed to perform mathematical modeling and statistical analysis.

#### Expected Learning Outcomes

• the ability to differentiate functions, find the limits using derivatives (compute limits by L’Hopital’s rule), study the functions and sketch their graphs using derivatives
• the knowledge of basic notions of set theory: sets, functions, basic elementary functions, ability to sketch their graphs using basic substitutions
• be able to compute derivatives of complex functions, to conduct function research and make a graph
• know the basic concepts and properties of matrices
• be able to calculate the determinant and apply its properties
• be able to explore system of linear algebraic equations using Cramer's formulas
• be able to solve linear algebraic equations system by Gauss method
• be able to use the Kronecker-Capelli (Rouché–Capelli) theorem correctly
• know the fundamentals of vector analysis and analytical Geometry
• be able to perform operations on vectors in coordinate form, solve problems in analytic geometry on a plane and in space
• be able to find a basis of a finite-dimensional linear space / subspace, decompose a vector in a basis
• to apply inner product, cross product and their properties
• know the definitions of inner product, cross product and triple product
• know the canonical, general and parametric equations of lines and planes
• identify relative position of straight line and plane
• demonstrate the knowledge of the notions of limit of a function, continuity
• ability to compute the limits and tell whether a function is continuous
• be able to calculate indefinite and definite integrals
• to use integrals in applications
• be able to calculate partial derivatives
• be able to calculate the limits of numerical sequences and functions
• be able to define the relevant sample space, compute the probabilities of random events
• be able to solve problems about random variables and their characteristics
• be able to solve the problem of finding the probabilities of random events and their characteristics
• be able to use Chebyshev inequality and Markov inequality
• be able to compute sample characteristics, construct the empirical distribution function, histogram and the frequency polygon
• be able to construct confidence intervals for the parameters of the normal distribution
• be able to test the hypothesis about the mean for samples from the normal distribution

#### Course Contents

• Elements of Linear Algebra
• Elements of vector algebra and analytic geometry
• Limits and continuity
• Basics of Differential Calculus. Applications
• Basics of Integral Calculus
• Functions of two variables
• Probability spaces
• Random variables
• Statistical hypothesis testing

#### Assessment Elements

• Test 1
The test is carried out in the classroom, in writing, 80 minutes. In the case of a distance learning format, control work is carried out remotely. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online).
• Test 2
The test is carried out in the classroom, in writing, 80 minutes. In the case of a distance learning format, control work is carried out remotely. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online).
• Classroom work 1
Classroom work is an aggregate indicator that includes the evaluation of students' work during practical seminars, scores for individual homework, and short quizzes in the classroom
• Classroom work 2
Classroom work is an aggregate indicator that includes the evaluation of students' work during practical seminars, scores for individual homework, and short quizzes in the classroom
• Classroom work 3
Classroom work is an aggregate indicator that includes the evaluation of students' work during practical seminars, scores for individual homework, and short quizzes in the classroom
• Exam
The test is carried out in the classroom, in writing, 80 minutes. In the case of a distance learning format, control work is carried out remotely. In the latter case, it can consist of two parts - written (extracurricular) and oral (classroom, including online). A student is eligible for credit for the final cumulative grade as a final summative grade if all tests are graded 8, 9, or 10.

#### Interim Assessment

• 2022/2023 4th module
0.084 * Classroom work 3 + 0.18 * Test 2 + 0.4 * Exam + 0.078 * Classroom work 2 + 0.18 * Test 1 + 0.078 * Classroom work 1

#### Recommended Core Bibliography

• Frederick J Gravetter, Larry B. Wallnau, Lori-Ann B. Forzano, & James E. Witnauer. (2020). Essentials of Statistics for the Behavioral Sciences, Edition 10. Cengage Learning.
• Hilbert, S. (2010). Calculus : An Active Approach with Projects. Washington, DC: Mathematical Association of America. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=490207
• Mangatiana A. Robdera, A Concise Approach to Mathematical Analysis, 2003, [electronic resource] link: https://link.springer.com/book/10.1007/978-0-85729-347-3
• Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics: Global Edition (Vol. Eight edition). Boston, Massachusetts: Pearson Education. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1417883
• V. A. Zorich. (2016). Mathematical Analysis I (Vol. 2nd ed. 2015). Springer.
• Williams, G. (2019). Linear Algebra with Applications (Vol. Ninth edition). Burlington, MA: Jones & Bartlett Learning. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1708709

#### Recommended Additional Bibliography

• C. R. J. Clapham, Introduction to Mathematical Analysis, 1973, Routledge & Kegan Paul. ISBN: 978-94-011-6572-3. [electronic resource] link: https://link.springer.com/book/10.1007/978-94-011-6572-3
• Fuad Aleskerov, Hasan Ersel, & Dmitri Piontkovski. (2011). Linear Algebra for Economists. Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.spr.sptbec.978.3.642.20570.5
• Ross, S. M. (2009). Introduction to Probability and Statistics for Engineers and Scientists (Vol. 4th ed). Burlington: Elsevier Ltd. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=414356
• Бурмистрова, Е. Б.  Линейная алгебра : учебник и практикум для академического бакалавриата / Е. Б. Бурмистрова, С. Г. Лобанов. — Москва : Издательство Юрайт, 2019. — 421 с. — (Бакалавр. Академический курс). — ISBN 978-5-9916-3588-2. — Текст : электронный // Образовательная платформа Юрайт [сайт]. — URL: https://urait.ru/bcode/425852 (дата обращения: 28.08.2023).
• Ковалев, Е. А.  Теория вероятностей и математическая статистика для экономистов : учебник и практикум для вузов / Е. А. Ковалев, Г. А. Медведев ; под общей редакцией Г. А. Медведева. — 2-е изд., испр. и доп. — Москва : Издательство Юрайт, 2020. — 284 с. — (Высшее образование). — ISBN 978-5-534-01082-4. — Текст : электронный // Образовательная платформа Юрайт [сайт]. — URL: https://urait.ru/bcode/450466 (дата обращения: 28.08.2023).
• Кремер, Н. Ш.  Теория вероятностей и математическая статистика : учебник и практикум для вузов / Н. Ш. Кремер. — 5-е изд., перераб. и доп. — Москва : Издательство Юрайт, 2019. — 538 с. — (Высшее образование). — ISBN 978-5-534-10004-4. — Текст : электронный // Образовательная платформа Юрайт [сайт]. — URL: https://urait.ru/bcode/431167 (дата обращения: 28.08.2023).
• Кудрявцев Л.Д. - КУРС МАТЕМАТИЧЕСКОГО АНАЛИЗА В 3 Т. ТОМ 2 В 2 КНИГАХ 6-е изд., пер. и доп. Учебник для бакалавров - М.:Издательство Юрайт - 2016 - 720с. - ISBN: 978-5-9916-6126-3 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/kurs-matematicheskogo-analiza-v-3-t-tom-2-v-2-knigah-387530
• Линейная алгебра и аналитическая геометрия. Практикум: Учебное пособие / А.С. Бортаковский, А.В. Пантелеев. - М.: НИЦ ИНФРА-М, 2015. - 352 с.: 60x90 1/16. - (Высшее образование: Бакалавриат). (переплет) ISBN 978-5-16-010206-1 - Режим доступа: http://znanium.com/catalog/product/476097
• Основы линейной алгебры и аналитической геометрии: Учебно-методическое пособие / В.Г. Шершнев. - М.: НИЦ ИНФРА-М, 2013. - 168 с.: 60x88 1/16. - (Высшее образование: Бакалавриат). (обложка) ISBN 978-5-16-005479-7 - Режим доступа: http://znanium.com/catalog/product/318084
• Путко Б.А., Тришин И.М., Кремер Н.Ш. - под ред. - МАТЕМАТИЧЕСКИЙ АНАЛИЗ В 2 Т. Учебник и практикум для академического бакалавриата - М.:Издательство Юрайт - 2016 - 634с. - ISBN: 978-5-9916-6238-3 - Текст электронный // ЭБС ЮРАЙТ - URL: https://urait.ru/book/matematicheskiy-analiz-v-2-t-388079