Course detail

# Statistics and Probability

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.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Entry knowledge

Foundations of differential and integral calculus.

Foundations of descriptive statistics, probability theory and mathematical statistics.

Rules for evaluation and completion of the course

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.

Aims

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

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

ANDĚL, Jiří. Základy matematické statistiky. 3., opr. vyd. Praha: Matfyzpress, 2011. ISBN 978-80-7378-001-2. (CS)

FELLER, W.: An Introduction to Probability Theory and its Applications. J. Wiley, New York 1957. ISBN 99-00-00147-X
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
Anděl, Jiří. Základy matematické statistiky. 3.,  Praha: Matfyzpress, 2011. ISBN 978-80-7378-001-2.
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
Meloun M., Militký J.: Statistické zpracování experimentálních dat (nakladatelství PLUS, 1994).

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Classification of course in study plans

• Programme MITAI Master's

specialization NSPE , 1 year of study, winter semester, compulsory
specialization NBIO , 1 year of study, winter semester, compulsory
specialization NSEN , 1 year of study, winter semester, compulsory
specialization NVIZ , 1 year of study, winter semester, compulsory
specialization NGRI , 1 year of study, winter semester, compulsory
specialization NADE , 1 year of study, winter semester, compulsory
specialization NISD , 1 year of study, winter semester, compulsory
specialization NMAT , 1 year of study, winter semester, compulsory
specialization NSEC , 1 year of study, winter semester, compulsory
specialization NISY up to 2020/21 , 1 year of study, winter semester, compulsory
specialization NCPS , 1 year of study, winter semester, compulsory
specialization NHPC , 1 year of study, winter semester, compulsory
specialization NNET , 1 year of study, winter semester, compulsory
specialization NMAL , 1 year of study, winter semester, compulsory
specialization NVER , 1 year of study, winter semester, compulsory
specialization NIDE , 1 year of study, winter semester, compulsory
specialization NEMB , 1 year of study, winter semester, compulsory
specialization NISY , 1 year of study, winter semester, compulsory
specialization NEMB up to 2021/22 , 1 year of study, winter semester, compulsory

#### Type of course unit

Lecture

26 hod., optionally

Teacher / Lecturer

Syllabus

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. 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.
5. Estimation of parameters using the method of moments and the maximum likelihood method.
6. Bayesian approach and construction of Bayesian estimates.
7. Extension of hypothesis tests for binomial and normal distributions.
8. Analysis of variance (simple sorting, ANOVA), post hos analysis.
9. Distribution tests.
10. Nonparametric methods of testing statistical hypotheses.
11. Regression analysis. Linear regression models. Testing hypotheses about regression model parameters.
12. Regression analysis. Comparison of regression models. Diagnostics. Nonlinear regression models.
13. Analysis of categorical data. Contingency table. Independence test. Four-field tables. Fisher's exact test.

Fundamentals seminar

23 hod., compulsory

Teacher / Lecturer

Syllabus

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. Reminder of discussed examples in the IPT subjekt
5. The method of moments and the maximum likelihood method.
6. Bayesian estimates.
7. Hypothesis tests for binomial and normal distributions.
8. Analysis of variance, post host analysis.
9. Tests on distribution, tests of good agreement.
10. Nonparametric methods of testing statistical hypotheses - part 1.
11. Regression analysis – linear regression models
12. Regression analysis – diagnostics, non-linear regression models
13. Analysis of categorical data. Contingency table. Four-field tables

Project

16 hod., compulsory

Teacher / Lecturer

Syllabus

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

Seminar

4 hod., optionally

Teacher / Lecturer

Syllabus

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

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