Course detail

Statistics and Probability

FIT-MSPAcad. year: 2019/2020

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

doc. RNDr. Libor Žák, Ph.D.

Department

Dept. of Statistics and Optimization (ÚM OSO)

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.

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.

Course curriculum

Not applicable.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Anděl, Jiří. Základy matematické statistiky. 3.,  Praha: Matfyzpress, 2011. ISBN 978-80-7378-001-2.
D. P. Bertsekas, J. N. Tsitsiklis. Introduction to Probability, Athena, 2008. Scientific
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, Karel. Regrese. 1., Praha: Matfyzpress, 2008. ISBN 978-80-7378-041-8

Classification of course in study plans

  • Programme MITAI Master's

    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 NISD , 1 year of study, winter semester, compulsory
    specialization NSEC , 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 NSPE , 1 year of study, winter semester, compulsory
    specialization NADE , 1 year of study, winter semester, compulsory
    specialization NMAT , 1 year of study, winter semester, compulsory
    specialization NISY , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Libor Žák, Ph.D.
doc. RNDr. Milan Češka, Ph.D.

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 hod., compulsory

Teacher / Lecturer

Ing. Kristína Šramková
prof. Ing. Tomáš Vojnar, Ph.D.
Ing. Pavel Hrabec, Ph.D.
doc. RNDr. Libor Žák, Ph.D.
doc. RNDr. Milan Češka, Ph.D.
doc. Mgr. Lukáš Holík, Ph.D.
doc. Mgr. Adam Rogalewicz, Ph.D.
Ing. Ondřej Lengál, Ph.D.

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 hod., compulsory

Teacher / Lecturer

Ing. Pavel Hrabec, Ph.D.
doc. RNDr. Milan Češka, Ph.D.
Ing. Ondřej Lengál, Ph.D.
prof. Ing. Tomáš Vojnar, Ph.D.
doc. Mgr. Adam Rogalewicz, Ph.D.
doc. Mgr. Lukáš Holík, Ph.D.
doc. RNDr. Libor Žák, Ph.D.
Ing. Kristína Šramková

Syllabus

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