Selected statistical Methods

Course guarantor:

Pavel Hrabák

Lecturer:

Pavel Hrabák

Tutor:

Pavel Hrabák, Jitka Hrabáková, Petr Novák, Jana Vacková

Supervisor:

Department of Applied Mathematics

Synopsis:

The course leads the student through advanced probabilistic and statistical methods used in information technology praxis. Particularly it deals with multivariate normal distribution, application of entropy in coding theory, hypothesis testing (T-tests, goodness of fit tests, independence test). Second part of the course deals with random processes with focus on Markov chains. The high point of the course is the Queuing theory and its application in networks.

Requirements:

Basics of probability and statistics, multivariable calculus, and linear algebra.

Syllabus of lectures:

1. Summary of basic terms of probability theory

2. Random variables

3. Random vectors

4. Multivariate normal distribution

5. Entropy for discrete distribution

6. Application of entropy in coding theory

7. Entropy of continuous distribution

8. Summary of basic terms of statistics

9. Paired and Two-sample T-test,

10. Goodness of fit tests,

11. Independence test, contingency table

12. Estimation od PDF and CDF

13. Gaussian mixtures and EM algorithm

14. Random processes - stacionarity

15. Random processes - examples (Gaussian, Poisson)

16. Memory-less distributions, exponential race

17. Markov chain with discrete time

18. Markov chain with discrete time - state classiffication

19. Markov chain with discrete time - stationarity

20. Markov chain with discrete time - parameters estimation

21 MCMC

22. Markov chain with continuous time

23. Markov chain with continuous time - Kolmogorov equations

24. Queuing theory, Little theorem

25. Queuing systems M/M/1 and M/M/m

26. Queuing systems M/G/infty

Syllabus of tutorials:

1. Revision lesson: basics of probability

2. Random vectors, multivariate normal distribution

3. Entropy and coding theory

4. Entropy, mutual information

5. T-tests

6. Goodness of fit tests, sndependence test

7. Estimation od PDF and CDF

8. Random processes, Poisson

9. Markov chain with discrete time - stationarity

10. Markov chain with discrete time - state classiffication

11. Exponential race

12. Markov chain with continuous time

13. Queuing theory

Study Objective:

The goal of the course is to introduce to the students advanced probabilistic and statistical methods used in information technology praxis.

Study materials:

1. Cover, T. M. - Thomas, J. A. : Elements of Information Theory (2nd Edition). Wiley, 2006. ISBN 978-0-471-24195-9.

2. Durrett, R. : Essentials of Stochastic Processes. Springer, 1999. ISBN 978-0387988368.

3. Grimmett, G. - Stirzaker, D. : Probability and Random Processes (3rd Edition). Oxford University Press Inc., 2001. ISBN 978-0-19-857222-0.

Note:

Further information:

https://courses.fit.cvut.cz/NI-VSM/

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Master specialization Computer Security, in Czech, 2020 (compulsory course in the program)
• Master specialization Design and Programming of Embedded Systems, in Czech, 2020 (compulsory course in the program)
• Master specialization Computer Systems and Networks, in Czech, 202 (compulsory course in the program)
• Master specialization Management Informatics, in Czech, 2020 (compulsory course in the program)
• Master specialization Software Engineering, in Czech, 2020 (compulsory course in the program)
• Master specialization System Programming, in Czech, version from 2020 (compulsory course in the program)
• Master specialization Web Engineering, in Czech, 2020 (compulsory course in the program)
• Master specialization Knowledge Engineering, in Czech, 2020 (compulsory course in the program)
• Master specialization Computer Science, in Czech, 2020 (compulsory course in the program)
• Mgr. programme, for the phase of study without specialisation, ver. for 2020 and higher (compulsory course in the program)
• Master specialization System Programming, in Czech, version from 2023 (compulsory course in the program)
• Master specialization Computer Science, in Czech, 2023 (compulsory course in the program)