Course details

# Statistics and Probability

MSP Acad. year 2019/2020 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.

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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 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.
- Nonparametric methods for testing statistical hypotheses.
- Analysis of categorical data: contingency table, chi-square test, Fisher test.
- Markov processes with discrete and continuous time and their analysis and applications.
- Markov decision processes and their analysis. Hidden Markov Models
- Introduction to randomized algorithms and their use (Monte Carlo, Las Vegas, applications).

Syllabus of numerical exercises

- Probability theory repetition: probability, conditioned probability, dependent and independent events, Bayes formula.
- Random variables repetition: discrete and continuous random variables, functional and numerical characteristics of random variable, basic discrete and continuous distributions.
- Random vector repetition: functions and numerical characteristics, distribution. Multidimensional normal distribution.
- Limit theorems and their use.
- 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. Diagnostics.
- Nonparametric methods for testing statistical hypotheses.
- Analysis of categorical data: contingency table, chi-square test.
- Markov processes with discrete and continuous time and their analysis and applications.
- Markov decision processes and their analysis.
- Introduction to randomized algorithms

Progress assessment

Two tests will be written during the semester - 6th and 12th week. The exact term will be specified by the instructor. The test duration is 45 minutes. The rating of each test is 0-15 points.

**Final written exam - 70 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. The points earned in the exercise are transferred to the exam.

Schedule

Day | Type | Weeks | Room | Start | End | Lect.grp | Groups | Info |
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Tue | lecture | lectures | D105 | 10:00 | 11:50 | 1MIT 2MIT | xx | |

Tue | exercise | lectures | A113 | 12:00 | 14:50 | 1MIT 2MIT | xx | |

Wed | exercise | lectures | A113 | 08:00 | 10:50 | 1MIT 2MIT | xx | |

Wed | exercise | lectures | A113 | 11:00 | 13:50 | 1MIT 2MIT | xx | |

Thu | exercise | lectures | D0207 | 08:00 | 10:50 | 1MIT 2MIT | xx | |

Thu | exercise | lectures | D0207 | 11:00 | 13:50 | 1MIT 2MIT | xx | |

Fri | exercise | lectures | D0207 | 08:00 | 10:50 | 1MIT 2MIT | xx | |

Fri | exercise | lectures | D0207 | 11:00 | 13:50 | 1MIT 2MIT | xx |

Course inclusion in study plans