Delivered at:: Bachelor's Programme in Digital Product Management

Course type:: Compulsory course

When:: 1 year, 4 module

Instructors

Gladkova, Margarita

Karpov, Gleb

Full Syllabus

Abstract

Probability Theory and Mathematical Statistics is a core mathematical subject taught to the second year students in the 1st and 2nd academic modules. The material is split between probability theory and statistics almost evenly. The course covers classical probability topics from basic probability to limit theorems. More attention is paid to the conditional moments of multivariate random variables. Depending on the available time more advanced topics such as random walks, the Poisson process and Markov chains may be considered. The statistical section starts with the descriptive techniques but quickly switches to the inferential methods as they are more mathematically involved and require more eorts to explain. The topics covered here include sampling distributions, point and interval estimates, hypothesis testing. We conclude with a univariate and, if time permits, multivariate regression. Throughout the course a certain balance between mathematical rigor and clarity is maintained. Sometimes this dilemma is resolved in favor of illustrative examples which help students capture the main ideas and use them in practice rather than focus on blind memorizing the derivations. However, we find it instructive to provide the tractable proofs whenever it makes pedagogical or some other sense. The course is taught in English and worth 5 credits.

Learning Objectives

Probability theory and Statistics provides an essential basis in probability theory and mathematical statistics and educates students how to use these principles in practice. Another implicit purpose of PT&Stat is to guide the rst steps in students' research by suggesting more challenging topics and problems to the interested students. This kind of activity develops self-study skills and critical thinking, emphasizes importance of working with literature and provides many more helpful skills

Expected Learning Outcomes

Be able to build, assess the quality and make predictions with the linear regression model.
Be able to derive and use the main sampling distributions - for mean, proportion and variance
Be able to estimate the main characteristics of a random variable by means of point or interval estimates. Know how to construct and interpret the confidence intervals for mean, proportion and variance.
Be able to use the concept of conditional probability, law of total probability, notion of independence, collectively exhaustive events.
Be able to use the methods of descriptive statistics to summarize and visualize the raw data.
Be aware of different definitions of probability, the axioms of probability and their use for derivation of major probabilistic relationships. Know the basic counting methods and principles of combinatorics.
Know and be able to use alternative ways of describing a continuous random variable - probability density function and cumulative distribution function. Know how to calculate basic characteristics of a continuous random variable. Be aware of commonly used continuous distributions.
Know how to work with a multivariate random variable using the joint probability distribution. Be able to detect the indepedent random variables, calculate the marginal and conditional distributions, covariance and correlation between the variables.
Understand the principles of hypothesis testing. Be able to perform tests for population mean, proportion and variance.
Understand the role of limit theorems in the probability theory. Be able to use the Markov and Chebyshev inequalities in practice. Be able to implement the central limit theorem.
Understand what is meant by a random variable. Know how to use the probability mass functions for calculating the basic characteristics of a discrete random variable (expected value, variance). Be aware of commonly used discrete distributions.

Course Contents

Axioms of probability
Jointly distributed random variables
Conditional probability and independence
Methods of descriptive statistics
Discrete random variables
Ideas of sampling and sampling distributions
Continuous random variables
Point and interval estimates
Limit theorems
Linear regression model
Hypothesis testing

Assessment Elements

Exam
Hometasks
Quizzes
Project
Exam
Hometasks
In-class tests
Quizz

Interim Assessment

2021/2022 4th module
0.2 * In-class tests + 0.1 * Quizz + 0.2 * Hometasks + 0.5 * Exam
2022/2023 1st module
0.2 * Hometasks + 0.1 * Quizzes + 0.45 * Exam + 0.25 * Project

Bibliography

Recommended Core Bibliography

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
Blitzstein, J. K., & Hwang, J. (2019). Introduction to Probability, Second Edition (Vol. Second edition). Boca Raton, FL: Chapman and Hall/CRC. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=nlebk&AN=2024519

Recommended Additional Bibliography

Balakrishnan, N., Koutras, M. V., & Konstantinos, P. (2019). Introduction to Probability : Models and Applications. Hoboken, NJ: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=2097342
Ghahramani, S. (2018). Fundamentals of Probability : With Stochastic Processes (Vol. Fourth edition). Boca Raton, FL: Chapman and Hall/CRC. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1875108
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
Linton, O. B. (2017). Probability, Statistics and Econometrics. London, United Kingdom: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1200673
Samuel Goldberg. (2013). Probability : An Introduction. [N.p.]: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1152975

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Probability Theory and Mathematical Statistics