Statistics and Probability

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

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

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:

• parametric estimation based on known probability distribution
• simultaneous testing of multiple parameters
• goodness of fit tests
• regression analysis including regression modeling
• nonparametric methods
• maximum likelihood estimation
• Markov processes
• randomised algorithms 

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
• Meloun M., Militký J.: Statistické zpracování experimentálních dat (nakladatelství PLUS, 1994).
• D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena, 2008. Scientific

• DeGroot, Morris H., Schervish, Mark J. Probability and Statistics (4th Edition). Boston: Addison-Wesley, 2010. ISBN 0-321-50046-61.

1. Summary and recall of knowledge and methods used in the subject of IPT - probability, random variable. Markov processes and their analysis.
2. Markov decision processes and their basic analysis.
3. Introduction to randomized algorithms and their use (Monte Carlo, Las Vegas, applications).
4. Selected probability distributions, conditional probability, likelihood and likelihood function
5. Construction of maximum likelihood estimators (MLE), and properties of MLE
6. Sufficient statistics, Fisher information and construction of asymptotic confidence intervals
7. Probability distribution of sums (averages) of selected random variables, classical construction of confidence intervals and introduction to hypothesis testing
8. Linear model, t-test for equal variances, ANOVA
9. Post-hoc comparisons for ANOVA, test for equal variances, normality test.
10. Linear regression model, tests for statistical significance of coefficients and submodels, confidence and prediction intervals for response
11. Linear regression model diagnostics.
12. Nonparametric hypotheses testing ans tests for categorical data..
13. Goodness of fit tests, likelihood ratio test, introduction to generalized linear models.

1. Application of basic statistical methods, statistic a programming.
2. Application of advanced statistical methods.

1. Application and analysis of Markov processes.
2. Basic application and analysis of Markov decision processes.
3. Design and analysis of basic randomised algorithms.
4. Condional probability, likelihood function.
5. Maximum likelihood estimation.
6. Sufficient statistics, Fisher information, asymptotical confidence intervals
7. Basic statistical hypothesis testing
8. ANOVA
9. Post-hoc comparisons, tests for equal variances, normality test.
10. Computation of linear regression model, tests for statistical significance of its coefficients and submodels.
11. Confidence anf prediction intervals for response in linear regression, regression diagnostics.
12. Selected nonparametric hypotheses tests, tests for categorical data.
13. Kolmogorov-Smirnov (Liliefors) test, likelihood ratio test.

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

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 10th week focuses on maximum likelihood estimation and basic hypotheses testing.

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.