Statistics and Probability

• 26 hrs lectures
• 34 hrs exercises
• 5 hrs projects

• 60 pts final exam (written part)
• 30 pts mid-term test (written part)
• 10 pts projects

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
• Markov processes

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.

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.

Foundations of differential and integral calculus.

Foundations of descriptive statistics, probability theory and mathematical statistics.

• 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
• D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena, 2008. Scientific

1. Summary of basic theory of probability: axiomatic definition of probability, conditioned probability, dependent and independent events, Bayes formula.
2. Summary of discrete and continuous random variables: probability, probability distribution density, distribution function and their properties, functional and numerical characteristics of random variable, basic discrete and  continuous distributions.
3. Discrete and continuous random vector (distribution functions, characteristics, multidimensional distribution). Transformation of random variables. Multidimensional normal distribution.
4. Limit theorems and their use (Markov and Chebyshev Inequalities, Convergence, Law of Large Numbers, Central Limit Theorem)
5. Parameter estimation. Unbiased and consistent estimates. Method of moments, Maximum likelihood method, Bayesian approach - parameter estimates.
6. Analysis of variance (simple sorting, ANOVA). Multiple comparison (Scheffy and Tukey methods).
7. Testing statistical hypotheses on distributions. Goodness of fit tests.
8. Regression analysis. Creating a regression model. Test hypotheses on regression model parameters. Comparison of regression models. Diagnostics.
9. Project assignment, demonstration of programs and tools for solving statistical problems.
10. Nonparametric methods for testing statistical hypotheses.
11. Analysis of categorical data: contingency table, chi-square test, Fisher test.
12. Markov processes, Markov decision processes, and their analysis and applications.
13. Introduction to randomized algorithms and their use (Monte Carlo, Las Vegas, applications).

1. Sets, relations, and their basic properties.
2. Propositional calculus and its formal system.
3. Repetition of the basic probability theory and statistics.
4. Important distribution and  their use in Limit theorems.
5. Parameter estimate: properties, methods
6. Analysis of variance (simple sorting, ANOVA), post hos analysis.
7. Testing statistical hypotheses on distributions. Goodness of fit tests.
8. Regression analysis. Creating a regression model. Test hypotheses on regression model parameters.
9. Regression analysis. Test hypotheses on regression model parameters. Diagnostics.
10. Nonparametric methods for testing statistical hypotheses.
11. Analysis of categorical data: contingency table, chi-square test.
12. Application and analysis of Markov processes and Markov decision processes.
13. Introduction to randomized algorithms

1.  Usage of tools for solving statistical problems (data processing and interpretation).

Three tests will be written during the semester - 3rd, 6th and 11th week. The exact term will be specified by the lecturer. The test duration is 60 minutes. The evaluation of each test is 0-10 points.

Projected evaluated 0-10 points.

Final written exam - 60 points
Exam prerequisites:

The credit will be awarded to the one who meets the attendance conditions and whose total test scores will reach at least 15 points and project score at least 5 points. The points earned in the exercise are transferred to the exam.

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.

The credit will be awarded to the one who meets the attendance conditions and whose total test scores will reach at least 15 points and project score at least 5 points. The points earned in the exercise are transferred to the exam.