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Regular version of the site
Master 2019/2020

Probability Theory and Mathematical Statistics

Category 'Best Course for Career Development'
Type: Bridging course (Data Science)
Area of studies: Applied Mathematics and Informatics
When: 1 year, 1 module
Mode of studies: offline
Master’s programme: Data Science
Language: English
ECTS credits: 3

Course Syllabus

Abstract

The overwhelming majority of courses on data-mining and artificial intelligence implies a strong background in probability and statistics. The goal of this course is to provide those students who are not so keen on the subject matter with its fundamentals.
Learning Objectives

Learning Objectives

  • To introduce the theoretical foundations of Probability theory.
  • To introduce the theoretical foundations of Mathematical statistics.
  • To provide the students with practical skills of modelling real-world in the framework of probability and statistics.
Expected Learning Outcomes

Expected Learning Outcomes

  • Understand problems of statistics.
  • Understand basic statistical models.
  • Understand Empirical distribution function, method of moments, maximum likelihood principle.
  • Understand fundamental concepts of multivariate normal distribution.
  • Understand fundamental concepts of statistical hypothesis.
  • Understand fundamental concepts of statistical hypothesis for the normal distribution.
  • Understand fundamental concepts chi-squared test.
  • Understand fundamental concepts, advantages and limitations of non-parametric statistics.
  • Understand tests for power-law distributions.
  • Understand fundamental concepts, advantages and limitations of Linear regression and Normal linear regression.
  • Solve simple problems using linear and normal linear regression.
Course Contents

Course Contents

  • Problems of Statistics. Parameter estimators. Basic statistical models. Distributions and sample statistics. Estimators. Unbiased estimators. Unbiased estimators for expectation and variance. Consistency. Efficiency and mean-squared error. Kramer-Rao inequality.
  • Empirical distribution function. Empirical parameters. Method of moments. Maximum likelihood principle. Likelihood and loglikelihood.
  • Multivariate normal distribution. Student and Fisher distributions. Normal sample.
  • Testing statistical hypothesis. Null hypothesis. Test statistics. Type I and type II errors. Significant level. Critical region and critical values.
  • Statistical hypothesis for the normal distribution. The one-sample t-test. Comparing two samples. Tests for variance.
  • Chi-squared test. For known parameters. For unknown parameters.
  • Non-parametric statistics. Sign test. Two-sided sign test. Wilcoxon sign-rank test.
  • Tests for power-law distributions. Hill test. Ibragimov’s corrections for confident intervals. Data-collapse approach. Clauset-Shalizi-Newman test.
  • Linear regression. Normal linear regression.
Assessment Elements

Assessment Elements

  • non-blocking Homework 1
  • non-blocking Homework 2
  • non-blocking Homework 3
  • non-blocking Homework 4
  • non-blocking Homework 5
  • non-blocking Homework 6
  • non-blocking Exam
Interim Assessment

Interim Assessment

  • Interim assessment (1 module)
    0.4 * Exam + 0.1 * Homework 1 + 0.1 * Homework 2 + 0.1 * Homework 3 + 0.1 * Homework 4 + 0.1 * Homework 5 + 0.1 * Homework 6
Bibliography

Bibliography

Recommended Core Bibliography

  • Dekking F. M. et al. A Modern Introduction to Probability and Statistics: Understanding why and how. – Springer Science & Business Media, 2005. – 488 pp.

Recommended Additional Bibliography

  • Wasserman, L. All of nonparametric statistics. – Springer Science & Business Media, 2006. – 270 pp.