Course details

# Statistics and Probability

MSP Acad. year 2020/2021 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

Deputy Guarantor

Language of instruction

Completion

Time span

Assessment points

Department

Lecturer

Instructor

Češka Milan, RNDr., Ph.D. (DITS FIT BUT)

Holík Lukáš, Mgr., Ph.D. (DITS FIT BUT)

Hrabec Pavel, Ing. (DM OSO FME BUT)

Lengál Ondřej, Ing., Ph.D. (DITS FIT BUT)

Rogalewicz Adam, doc. Mgr., Ph.D. (DITS FIT BUT)

Šramková Kristína, Ing. (FME BUT)

Vojnar Tomáš, prof. Ing., Ph.D. (DITS FIT BUT)

Žák Libor, Doc. RNDr., Ph.D. (DM OSO FME BUT)

Subject specific learning outcomes and competences

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

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.

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 kwnowledge and skills

Foundations of differential and integral calculus.

Foundations of descriptive statistics, probability theory and mathematical statistics.

Fundamental literature

- 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

Syllabus of lectures

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

Syllabus of numerical exercises

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

**Demo exercise focusing on algebra and logic (only the first two weeks -- 4-times 2 hours):**

- Sets, Cartesian product, relations, and functions. Properties and types of relations and functions. Congruence.
- Basic algebraic structures (group, Boolean algebra, lattice, field). Homomorfism.
- Propositional calculus. Syntax and semantics. Formal system for propositional calculus. Posts completeness theorem.
- Predicate logic. Syntax and semantics. Formal system for predicate logic. Gödels completeness theorem. Gödels incompleteness theorem.

Syllabus - others, projects and individual work of students

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

Progress assessment

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**

Controlled instruction

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

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.

Schedule

Day | Type | Weeks | Room | Start | End | Lect.grp | Groups | Info |
---|---|---|---|---|---|---|---|---|

Mon | exam | 2021-01-11 | A112 A113 D0206 D0207 D105 E104 E105 E112 G108 G202 M103 M104 M105 N103 N104 N105 N203 N204 N205 O204 | 10:00 | 12:50 | 1MIT 2MIT | řádná | |

Mon | exam | 2021-01-25 | C228 M103 M104 M105 N103 N203 N204 N205 | 11:00 | 13:50 | 1MIT | 1. oprava | |

Mon | exercise | lectures | D105 | 18:00 | 19:50 | 1MIT 2MIT | NBIO - NSPE xx | |

Tue | lecture | lectures | D105 | 10:00 | 11:50 | 1MIT 2MIT | NBIO - NSPE xx | doc. Žák |

Tue | lecture | 2., 3., 4., 5., 6., 7., 8., 10., 11., 12., 13. of lectures | D105v | 10:00 | 11:50 | YT, ZP, MST; Češka | ||

Tue | exercise | lectures | A113 | 12:00 | 13:50 | 1MIT 2MIT | xx | doc. Žák |

Tue | exercise | 3., 4., 5., 6., 7., 8., 10., 11., 12., 13. of lectures | A113v | 12:00 | 13:50 | TM, MST, Epson | ||

Wed | exercise | lectures | A113 | 08:00 | 09:50 | 1MIT 2MIT | xx | ing. Hrabec |

Wed | exercise | 3., 12., 13. of lectures | A113v | 08:00 | 09:50 | TM, MST, Epson | ||

Wed | exercise | lectures | A113 | 10:00 | 11:50 | 1MIT 2MIT | xx | ing. Hrabec |

Wed | exercise | 3., 12., 13. of lectures | A113v | 10:00 | 11:50 | TM, MST, Epson | ||

Thu | exercise | lectures | D0207 | 08:00 | 09:50 | 1MIT 2MIT | xx | ing. Šramková |

Thu | exercise | lectures | D0207v | 08:00 | 09:50 | TM, MST, Epson | ||

Thu | exercise | lectures | D0207 | 10:00 | 11:50 | 1MIT 2MIT | xx | ing. Šramková |

Thu | exercise | lectures | D0207v | 10:00 | 11:50 | TM, MST, Epson | ||

Thu | exercise | lectures | E104 E105 E112 | 18:00 | 19:50 | 1MIT 2MIT | NBIO - NSPE xx | |

Thu | exercise | 2. of lectures | E112v | 18:00 | 19:50 | YT, ZP | ||

Fri | exercise | lectures | D0207 | 08:00 | 09:50 | 1MIT 2MIT | xx | ing. Šramková |

Fri | exercise | lectures | D0207v | 08:00 | 09:50 | TM, MST, Epson | ||

Fri | exercise | lectures | D0207 | 10:00 | 11:50 | 1MIT 2MIT | xx | ing. Šramková |

Fri | exercise | lectures | D0207v | 10:00 | 11:50 | TM, MST, Epson | ||

Fri | exam | 2021-02-05 | A112 A113 D0206 D0207 D105 E104 E105 E112 G202 | 11:00 | 13:50 | 1MIT 2MIT | 2. oprava | |

Fri | exercise | 2020-09-25 | D105 | 16:00 | 17:50 | 1MIT |

Course inclusion in study plans