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# Probability Theory and Mathematical Statistics

2018/2019
ENG
Instruction in English
6
ECTS credits

#### Authors

Litvinova, Viktoria

Razgulyaeva, Liudmila
Course type:
Compulsory course
When:
1 year, 3, 4 module

#### Instructors

Nikitin, Yakov Y.

### Course Syllabus

#### Abstract

Probability and 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, ideas and techniques needed to solve simple problems of Probability and to perform the statistical Data Analysis. The first module introduces the basic ideas and initial knowledge in Probability theory. The second module deals with the theory of random variables and Descriptive Statistics. In the third module the students will learn the basic notions of Mathematical Statistics.

#### Learning Objectives

• to introduce the students to the basic mathematical notions, ideas and techniques needed to solve simple problems of Probability and to perform the statistical Data Analysis

#### Expected Learning Outcomes

• Is able to determine the events in question, solve problems of finding the probabilities of events
• Can solve problems with sequences of Bernoulli trials
• Is able to solve problems with random variables and their characteristics
• Is able to calculate the numerical characteristics of a random variable, knows its basic properties.
• Knows the basic laws of distribution of continuous and discrete random variables
• Is able to solve the problem of constructing confidence intervals for the parameters of the normal law; hypothesis testing of the average for normal samples.
• Can use point estimation and interval estimation
• Can test parametric and nonparametric hypotheses
• can conduct correlation and regression analysis

#### Course Contents

• Random events and Probability axioms
Random events. Axioms of probability, classical definition of probability, conditional probability, independent events, the law of total probability, Bayes’ rule
• Bernoulli trials
Sequences of Bernoulli trials
• Random variables and their description
Discreet random variables. Distribution functions of discreet random variables. Examples. Continuous random variables. Cumulative distribution function, probability density function. Examples
• Numerical characteristics of random variables
Expected value of a random variable, variance of a random variable
• Basic Laws of Probability
The Law of Large Numbers, Central Limit Theorem
• Statistical sample and its description
Descriptive statistics. Random samples and their main characteristics. Random samples from normal distribution
• Estimation theory: basic facts
Point estimation and interval estimation
• Statistical hypothesis testing
Testing of statistical hypothesis, chi-squared test
• Correlation and regression
Basics of regression analysis, least squares method, basics of correlation analysis

• Activity
• Test 1
• Test 2
• Exam

#### Interim Assessment

• Interim assessment (4 module)
0.18 * Activity + 0.55 * Exam + 0.112 * Test 1 + 0.158 * Test 2

#### Recommended Core Bibliography

• Deep, R. (2006). Probability and Statistics : With Integrated Software Routines. Amsterdam: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=196153
• Young, G. A., & Smith, R. L. (2005). Essentials of Statistical Inference. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=138968

#### Recommended Additional Bibliography

• Bruce, P. C. (2014). Introductory Statistics and Analytics : A Resampling Perspective. Hoboken, New Jersey: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=923330