# Subject description - B0B01PST1

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

 Page updated 19.5.2024 15:51:18, semester: Z/2024-5, Z,L/2023-4, Send comments about the content to the Administrators of the Academic Programs Proposal and Realization: I. Halaška (K336), J. Novák (K336)