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.

Learning outcomes of the course unit

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

Prerequisites

Foundations of differential and integral calculus.

Foundations of descriptive statistics, probability theory and mathematical statistics.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Anděl, Jiří. Základy matematické statistiky. 3.,  Praha: Matfyzpress, 2011. ISBN 978-80-7378-001-2.
Meloun M., Militký J.: Statistické zpracování experimentálních dat, 1994.
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 K.. Regresní analýza, Academia, Praha, 1989
D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena, 2008. Scientific

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

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.

Language of instruction

Czech

Work placements

Not applicable.

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.

Specification of controlled education, way of implementation and compensation for absences

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.

Classification of course in study plans

• Programme MITAI Master's

specialization NADE , 1. year of study, winter semester, 6 credits, compulsory
specialization NBIO , 1. year of study, winter semester, 6 credits, compulsory
specialization NGRI , 1. year of study, winter semester, 6 credits, compulsory
specialization NNET , 1. year of study, winter semester, 6 credits, compulsory
specialization NVIZ , 1. year of study, winter semester, 6 credits, compulsory
specialization NCPS , 1. year of study, winter semester, 6 credits, compulsory
specialization NSEC , 1. year of study, winter semester, 6 credits, compulsory
specialization NEMB , 1. year of study, winter semester, 6 credits, compulsory
specialization NHPC , 1. year of study, winter semester, 6 credits, compulsory
specialization NISD , 1. year of study, winter semester, 6 credits, compulsory
specialization NIDE , 1. year of study, winter semester, 6 credits, compulsory
specialization NISY , 1. year of study, winter semester, 6 credits, compulsory
specialization NMAL , 1. year of study, winter semester, 6 credits, compulsory
specialization NMAT , 1. year of study, winter semester, 6 credits, compulsory
specialization NSEN , 1. year of study, winter semester, 6 credits, compulsory
specialization NVER , 1. year of study, winter semester, 6 credits, compulsory
specialization NSPE , 1. year of study, winter semester, 6 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

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).

Fundamentals seminar

34 hours, compulsory

Teacher / Lecturer

Syllabus

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

Demo exercise focusing on algebra and logic (only the first two weeks -- 4-times 2 hours):
1. Sets, Cartesian product, relations, and functions. Properties and types of relations and functions. Congruence.
2. Basic algebraic structures (group, Boolean algebra, lattice, field). Homomorfism.
3. Propositional calculus. Syntax and semantics. Formal system for propositional calculus. Posts completeness theorem.
4. Predicate logic. Syntax and semantics. Formal system for predicate logic. Gödels completeness theorem. Gödels incompleteness theorem.

Project

5 hours, compulsory

Teacher / Lecturer

Syllabus

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

eLearning

eLearning: opened course