Probability and Statistics

Garant předmětu:

Pavel Hrabák

Lecturer:

Francesco Dolce, Pavel Hrabák

Tutor:

Francesco Dolce

Supervisor:

Department of Applied Mathematics

Synopsis:

Students will learn the basics of probabilistic thinking, the ability to synthesize prior and posterior information and learn to work with random variables. They will be able to apply basic models of random variable distributions and solve applied probabilistic problems in informatics and computer science. Using the statistical induction they will be able to perform estimations of unknown distributional parameters from random sample characteristics. They will also be introduced to the methods for testing statistical hypotheses and determining the statistical dependence of two or more random variables.

Requirements:

Basics of combinatorics and mathematical analysis.

Syllabus of lectures:

1. Probability - random events, event space structure, probability of a random event and its basic properties.

2. Conditional probability - dependent and independent events, Bayes theorem.

3. Random variables - distribution function of a random variable, continuous and discrete distributions, quantiles, median.

4. Characteristics of random variables - expected value, variance, general moments, kurtosis and skewness.

5. Overview of basic distributions - binomial, geometric, Poisson, uniform, normal, exponential. Their basic properties.

6. Random vectors - joint and marginal statistics, correlation coefficient, dependence and independence of random variables.

7. Random vectors - conditional distributions, sums of random variables.

8. Limit theorems - laws of large numbers, central limit theorem.

9. Statistical estimation - classification and processing of data sets, graphical representation of data, random sample, point estimation, basic sample statistics, sample mean and variance.

10. Interval estimation - confidence intervals for expectation and variance.

11. Hypothesis testing - testing strategy, tests for expectation and variance, their modifications.

12. Application of statistical testing in computer science.

13. Correlation and regression analysis: Linear and quadratic regression, sample correlation.

Syllabus of tutorials:

1. Basics of probability.

2. Conditional probability.

3. Random variables.

4. Basic characteristics of random variables.

5. Using basic distributions.

6. Random vectors - independence, covariance.

7. Random vectors - conditional distributions and sums.

8. Limit theorems

9. Processing of sets of data.

10. Statistical point estimation.

11. Interval estimation.

12. Hypotheses testing.

13. Regression and correlation analysis.

Study Objective:

The goal of the module is to introduce the students to basics of probability theory and mathematical statistics while focusing on applications in informatics.

Study materials:

1. Ahn H. : Probability and Statistics for Science and Engineering with Examples in R. Cognella, 2017. ISBN 978-1516513987.

2. Johnson J. L. : Probability and Statistics for Computer Science. Wiley-Interscience, 2008. ISBN 470383429.

3. Bonselet Ch. : Probability, Statistics, and Random Signals. Oxford University Press, 2016. ISBN 978-0190200510.

4. Grimmett G. R., Stirzaker D. R. : Probability and Random Processes (3rd Edition). Oxford University Press, 2001. ISBN 0-19-857223-9.

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Bachelor specialization, Computer Engineering, 2021 (compulsory course in the program)
• Bachelor specialization, Information Security, 2021 (compulsory course in the program)
• Bachelor specialization, Software Engineering, 2021 (compulsory course in the program)
• Bachelor specialization, Computer Science, 2021 (compulsory course in the program)
• Bachelor specialization, Computer Networks and Internet, 2021 (compulsory course in the program)
• Bachelor specialization Computer Systems and Virtualization, 2021 (compulsory course in the program)