Probability and Statistics

Garant předmětu:

Petr Hájek

Lecturer:

Kateřina Helisová

Tutor:

Dominik Beck, Kateřina Helisová, Matěj Lebeda, Matvei Slavenko

Supervisor:

Department of Mathematics

Synopsis:

Basics of probability theory and mathematical statistics. Includes descriptions of probability, random variables and their distributions, characteristics and operations with random variables. Basics of mathematical statistics: Point and interval estimates, methods of parameters estimation and hypotheses testing, least squares method. Basic notions and results of the theory of Markov chains.

Requirements:

Linear Algebra, Calculus, Discrete Mathematics

Syllabus of lectures:

1. Basic notions of probability theory. Kolmogorov model of probability. Independence, conditional probability, Bayes formula.

2. Random variables and their description. Random vector. Probability distribution function.

3. Quantile function. Mixture of random variables.

4. Characteristics of random variables and their properties. Operations with random variables. Basic types of distributions.

5. Characteristics of random vectors. Covariance, correlation. Chebyshev inequality. Law of large numbers. Central limit theorem.

6. Basic notions of statistics. Sample mean, sample variance. Interval estimates of mean and variance.

7. Method of moments, method of maximum likelihood. EM algorithm.

8. Hypotheses testing. Tests of mean and variance.

9. Goodness-of-fit tests.

10. Tests of correlation, non-parametic tests.

11. Discrete random processes. Stationary processes. Markov chains.

12. Classification of states of Markov chains.

13. Asymptotic properties of Markov chains. Overview of applications.

Syllabus of tutorials:

1. Elementary probability.

2. Kolmogorov model of probability. Independence, conditional probability, Bayes formula.

3. Mixture of random variables.

4. Mean. Unary operations with random variables.

5. Dispersion (variance). Random vector, joint distribution. Binary operations with random variables.

6. Sample mean, sample variance. Chebyshev inequality. Central limit theorem.

7. Interval estimates of mean and variance.

8. Method of moments, method of maximum likelihood.

9. Hypotheses testing. Goodness-of-fit tests.

10. Tests of correlation. Non-parametic tests.

11. Discrete random processes. Stationary processes. Markov chains.

12. Classification of states of Markov chains.

13. Asymptotic properties of Markov chains.

Study Objective:

Basics of probability theory and their application in statistical estimates and tests.

The use of Markov chains in modeling.

Study materials:

[1] Wasserman, L.: All of Statistics: A Concise Course in Statistical Inference. Springer Texts in Statistics, Corr. 2nd printing, 2004.

[2] Papoulis, A., Pillai, S.U.: Probability, Random Variables, and Stochastic Processes. McGraw-Hill, Boston, USA, 4th edition, 2002.

[3] Mood, A.M., Graybill, F.A., Boes, D.C.: Introduction to the Theory of Statistics. 3rd ed., McGraw-Hill, 1974.

Note:

Further information:

https://math.fel.cvut.cz/en/people/heliskat/01pst2.html

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomT4:D2-256 Helisová K. 14:30–16:00 (lecture parallel1) Dejvice Posluchárna 256
Tue	roomT2:C2-86 Helisová K. 11:00–12:30 (lecture parallel1 parallel nr.101) Dejvice CvičebnaroomT2:C2-86 Beck D. 14:30–16:00 (lecture parallel1 parallel nr.105) Dejvice Cvičebna roomT2:C3-52 Beck D. 12:45–14:15 (lecture parallel1 parallel nr.102) Dejvice T2:C3-52
Wed	roomT4:D2-256 Helisová K. 12:45–14:15 (lecture parallel1) Dejvice Posluchárna 256roomT2:C2-82 Helisová K. 14:30–16:00 (lecture parallel1 parallel nr.103) Dejvice T2:C2-82roomT2:C2-86 Helisová K. 16:15–17:45 (lecture parallel1 parallel nr.104) Dejvice Cvičebna
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Cybernetics and Robotics 2016 (compulsory course in the program)