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B0B01PST1	Probability and Statistics
Roles:	P	Extent of teaching:	4P+2S
Department:	13101	Language of teaching:
Guarantors:	Helisová K.	Completion:	Z,ZK
Lecturers:	Helisová K., Staněk J.	Credits:	6
Tutors:	Beck D., Helisová K., Staněk J.	Semester:	Z

Web page:

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

Anotation:

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.

Study targets:

Basics of probability theory and their application in statistical estimates and tests. The use of Markov chains in modeling.

Course outlines:

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.

Exercises outline:

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.

Literature:

[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.

Requirements:

Linear Algebra, Calculus, Discrete Mathematics

Note:

A necessary condition for the assignment is active participation at seminars and a successful test. More info: http://cmp.felk.cvut.cz/~navara/stat/

Keywords:

probability theory, statistical estimate, hypotheses testing, Markov chain

Subject is included into these academic programs:

Program	Branch	Role	Recommended semester
BPKYR_2021	Common courses	P	3

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Page updated 15.11.2025 17:51:52, semester: L/2026-7, Z/2025-6, L/2024-5, L/2025-6, Z/2026-7, Send comments about the content to the Administrators of the Academic Programs

Proposal and Realization: I. Halaška (K336), J. Novák (K336)