Course details

# Statistics and Probability

MSP Acad. year 2023/2024 Winter semester 6 credits

Summary of elementary concepts from probability theory and mathematical statistics. Limit theorems and their applications. Parameter estimate methods and their properties. Scattering analysis including post hoc analysis. Distribution tests, tests of good compliance, regression analysis, regression model diagnostics, non-parametric methods, categorical data analysis. Markov decision-making processes and their analysis, randomized algorithms.

Guarantor

Course coordinator

Language of instruction

Completion

Time span

- 26 hrs lectures
- 4 hrs seminar
- 23 hrs exercises
- 16 hrs projects

Assessment points

- 60 pts final exam (written part)
- 20 pts mid-term test (written part)
- 20 pts projects

Department

Lecturer

Češka Milan, doc. RNDr., Ph.D. (DITS)

Hrabec Pavel, Ing., Ph.D. (IM DSO)

Žák Libor, doc. RNDr., Ph.D. (IM DSO)

Instructor

Benko Matej, Ing. (IM)

Češka Milan, doc. RNDr., Ph.D. (DITS)

Eryganov Ivan, Ing., Ph.D. (IM DSO)

Hrabec Pavel, Ing., Ph.D. (IM DSO)

Macák Filip, Ing. (DITS)

Mrázek Vojtěch, Ing., Ph.D. (DCSY)

Žák Libor, doc. RNDr., Ph.D. (IM DSO)

Learning objectives

Introduction of further concepts, methods and algorithms of probability theory, descriptive and mathematical statistics. Development of probability and statistical topics from previous courses. Formation of a stochastic way of thinking leading to formulation of mathematical models with emphasis on information fields.

Students will extend their knowledge of probability and statistics, especially in the following areas:

- Parameter estimates for a specific distribution
- simultaneous testing of multiple parameters
- hypothesis testing on distributions
- regression analysis including regression modeling
- nonparametric methods
- creation of parameter estimates
- Bayesian statistics
- Markov processes
- randomised algorithms

Why is the course taught

The society development desires also technology and, in particular, information technology expansion. It is necessary to process information - data in order to control technology. Nowadays, there is a lot of devices that collect data automatically. So we have a large amount of data that needs to be processed. Statistical methods are one of the most important means of processing and sorting data, including their analysis. This allows us to obtain necessary information from your data to evaluate and control.

Prerequisite knowledge and skills

Foundations of differential and integral calculus.

Foundations of descriptive statistics, probability theory and mathematical statistics.

Study literature

- Anděl, Jiří. Základy matematické statistiky. 3., Praha: Matfyzpress, 2011. ISBN 978-80-7378-001-2.
- FELLER, W.: An Introduction to Probability Theory and its Applications. J. Wiley, New York 1957. ISBN 99-00-00147-X
- Hogg, V.R., McKean J.W. and Craig A.T. Introduction to Mathematical Statistics. Seventh Edition, 2012. Macmillan Publishing Co., INC. New York. ISBN-13: 978-0321795434 2013
- Zvára, Karel. Regrese. 1., Praha: Matfyzpress, 2008. ISBN 978-80-7378-041-8
- Meloun M., Militký J.: Statistické zpracování experimentálních dat (nakladatelství PLUS, 1994).
- D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena, 2008. Scientific

Syllabus of lectures

- Summary and recall of knowledge and methods used in the subject of IPT - probability, random variable. Markov processes and their analysis.
- Markov decision processes and their basic analysis.
- Introduction to randomized algorithms and their use (Monte Carlo, Las Vegas, applications).
- Summary and recall of knowledge and methods used in the subject of IPT (estimates, statistical tests). An outline of other areas of probability and statistics that will be covered.
- Estimation of parameters using the method of moments and the maximum likelihood method.
- Bayesian approach and construction of Bayesian estimates.
- Extension of hypothesis tests for binomial and normal distributions.
- Analysis of variance (simple sorting, ANOVA), post hos analysis.
- Distribution tests.
- Nonparametric methods of testing statistical hypotheses.
- Regression analysis. Linear regression models. Testing hypotheses about regression model parameters.
- Regression analysis. Comparison of regression models. Diagnostics. Nonlinear regression models.
- Analysis of categorical data. Contingency table. Independence test. Four-field tables. Fisher's exact test.

Syllabus of seminars

- Application of basic statistical methods, statistic a programming.
- Application of advanced statistical methods.

Syllabus of numerical exercises

- Application and analysis of Markov processes.
- Basic application and analysis of Markov decision processes.
- Design and analysis of basic randomised algorithms.
- Reminder of discussed examples in the IPT subjekt
- The method of moments and the maximum likelihood method.
- Bayesian estimates.
- Hypothesis tests for binomial and normal distributions.
- Analysis of variance, post host analysis.
- Tests on distribution, tests of good agreement.
- Nonparametric methods of testing statistical hypotheses - part 1.
- Regression analysis – linear regression models
- Regression analysis – diagnostics, non-linear regression models
- Analysis of categorical data. Contingency table. Four-field tables

Syllabus - others, projects and individual work of students

- Basic statistics and programming.
- Usage of tools for solving statistical problems (data processing and interpretation).

Progress assessment

The evaluation of the course consists of the test in the 5th week (max. 10 points) and the test in the 10th week (max. 10 points), the two projects (max 8 + 12 points), and the final exam (max 60 points).

The written test in the 5th week focuses on Markov processes and on basic randomized algorithms. The written test in the 9th week focuses on advance topics in statistics (will be clarified later).

Projects:

1st project: 8 points (2 points minimum) -- Statistics and programming.

2nd project: 12 points (4 points minimum) -- Advanced statistics.

**The requirements to obtain the accreditation that is required for the final exam: The minimal total score of 20 points achieved from the projects and from the tests in the 5th and 10th week (i.e. out of 40 points).**

**The final written exam: 0-60 points. Students have to achieve at least 25 points, otherwise the exam is assessed by 0 points.**

Participation in lectures in this subject is not controlled

Participation in the exercises **is compulsory**. During the semester two abstentions are tolerated. Replacement of missed lessons is determined by the leading exercises.

Exam prerequisites

Fulfil the attendance conditions and achieve in total at least 20 points from the tests and the projects.

Course inclusion in study plans