Course detail

Statistics and Probability

FIT-MSPAcad. year: 2024/2025

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.

Guarantor

Ing. Pavel Hrabec, Ph.D.

Department

Dept. of Statistics and Optimization (ÚM OSO)

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 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.

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:

  • 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 

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

Recommended reading

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).

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme MITAI Master's

    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 NNET , 1 year of study, winter semester, compulsory
    specialization NMAL , 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 NVER , 1 year of study, winter semester, compulsory
    specialization NIDE , 1 year of study, winter semester, compulsory
    specialization NISY , 1 year of study, winter semester, compulsory
    specialization NEMB , 1 year of study, winter semester, compulsory
    specialization NSPE , 1 year of study, winter semester, compulsory
    specialization NEMB , 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

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

Ing. Ivan Eryganov, Ph.D.
Ing. Matej Benko
Ing. Pavel Hrabec, Ph.D.
doc. RNDr. Libor Žák, Ph.D.
doc. RNDr. Milan Češka, Ph.D.

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. 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.

Seminar

4 hod., optionally

Teacher / Lecturer

Ing. Vojtěch Mrázek, Ph.D.
Ing. Pavel Hrabec, Ph.D.

Syllabus

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

Fundamentals seminar

23 hod., compulsory

Teacher / Lecturer

Ing. Pavel Hrabec, Ph.D.
Ing. Roman Andriushchenko
Ing. Matej Benko
doc. RNDr. Libor Žák, Ph.D.
Ing. Filip Macák
doc. RNDr. Milan Češka, Ph.D.
Ing. Ivan Eryganov, Ph.D.
Ing. David Hudák

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. 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.

Project

16 hod., compulsory

Teacher / Lecturer

Ing. Pavel Hrabec, Ph.D.
Ing. Matej Benko

Syllabus

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

Elearning

eLearning: currently opened course