Bachelor 2020/2021

Mathematics and Statistics

Type: Elective course (Political Science and World Politics)

Area of studies: Political Science

Delivered by: Department of Mathematics

Where: Saint-Petersburg School of Social Sciences

When: 1 year, 1-3 module

Mode of studies: offline

Instructors: Alexandra Grinikh, Yaroslavna Pankratova

Language: English

ECTS credits: 8

Contact hours: 92

Course Syllabus

Abstract

Mathematical Statistics has become an indispensable tool in almost every field of applied science, including social sciences. The goal of this course is to introduce the students to the basic mathematical notions and techniques needed to perform statistical analysis. The first module introduces the basic techniques of linear algebra and the coordinate method. The second module deals with the fundamentals of differential and integral calculus. In the third module the students will learn the basic notions of probability theory and mathematical statistics.

Learning Objectives

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

Expected Learning Outcomes

• can solve equations with one unknown
• can perform operations on vectors in coordinate form, solve problems in analytic geometry on a plane and in space
• can calculate the limits of numerical sequences and functions
• able to compute derivatives of complex functions, limits by rule of Lopital, to conduct function research and make a graph
• can calculate indefinite and definite integrals
• can calculate partial derivatives
• can solve the problem of finding the probabilities of random events
• can solve random value problems
• can test statistical hypotheses

Course Contents

• Elements of Linear Algebra
  Matrixes. Actions with matrices. Square matrix. Determinant. The inverse matrix. The rank of a matrix. Gaussian elimination method. Kramer's Theorem. Kronecker-Capelli Theorem.
• Elements of vector algebra and analytic geometry
  Vector space. Geometric interpretation of the vector. Linear operations on vectors and their properties. Collinear vectors. Coplanar vectors. Single orts. Scalar product of vectors and its properties. Length (norm) of the vector. The angle between the vectors. A linear combination of the vector system. Linear dependence and independence of vectors. Basis. Decomposition of the vector on the basis. Vector product and its properties. Mixed product of vectors. The equation of a line. Types of equations of a straight line the Distance from a point to a straight line. The angle between the lines, the condition of parallelism and perpendicularity. The equation of the circle. Plane in space, types of the equation of the plane. A straight line in space, Canonical and parametric equations of a straight line. Relative position of straight line and plane
• Limits and continuity
  Sequence, divergent sequence, limits of the sequence. Functions. Limits of the function. Properties of the function limits. Indeterminate forms. Fundamental limits. Equivalent functions. Continuity of the function. Discontinuity points and their classification
• Basics of Differential Calculus. Applications
  The definition of the derivative, its physical and geometrical sense. The relationship of continuity and differentiability. Derivative of sum, product and particular. Table of derivatives of basic elementary functions. Differential. Higher order derivatives and differentials. Some theorems on differentiable functions. L'hopital's Rule. Increasing and decreasing function. Extremum. Convexity, concavity, inflection points, asymptotes. Study of the function and its plotting.
• Basics of Integral Calculus
  An antiderivative and indefinite integral. Properties of the indefinite integral. Table of integrals. Basic methods of integration. The definition of a certain integral and its geometric meaning. Properties of a certain integral. Newton-Leibniz Formula
• Functions of two variables
  The definition of a function of two variables, domain, graph. Limit. Continuity. Partial derivatives of functions of several variables
• Probability spaces
  Random events. Actions with random events. Probability space. Classical definition of probability. Geometric probability. Conditional probability. The formula of total probability. Bayes formula. A sequence of independent Bernoulli trials. The most probable number of successes. A local limit theorem (de Moivre-Laplace). Integral limit theorem. Bernoulli's Theorem. Poisson's Theorem.
• Random variables
  Definition of a random variable. Distribution function. Discrete and continuous distributions. The distribution density of a random variable. Multidimensional distribution. Independence of random variables. Correlation coefficient. Chebyshev inequality. Markov Inequality.
• Statistical hypothesis testing
  Hypothesis testing. Errors of the first and second kind. Construction of confidence intervals for parameters of a normal distribution. Verification of parametric hypotheses. Testing hypotheses about parameters of a normal distribution (mean, variance). Testing hypotheses about the type of distribution.

Assessment Elements

• Homework
• Quizzes
• Test 1
• Test 2
• Test 3
• Exam

Interim Assessment

• Interim assessment (3 module)
  0.4 * Exam + 0.05 * Homework + 0.1 * Quizzes + 0.15 * Test 1 + 0.15 * Test 2 + 0.15 * Test 3

Bibliography

Recommended Core Bibliography

• 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

Recommended Additional Bibliography

• 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