Advanced Probability

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The subject is devoted to advanced Theory of probability and statistics on measure-theoretic level for general distributions of random variables. We deal with sample and integral characteristics of random variables and convergence criteria. Further, the theory of statistical model estimation and testing is extended for parametric and nonparametric cases.

Requirements:

Basic course of Calculus (in the extent of the courses 01MAA3-4 or 01MAB3-4 held at the FNSPE CTU in Prague), further 01PRST.

Syllabus of lectures:

Measurable spaces, sigma-fields, probability measures. Borel sets, measurable functions, random variables and probability distributions. Radon-Nikodym theorem. Product measure, integral w.r.t. probability measure. Expectation of random variables, Lp space. Characteristic function and its properties, applications. Kolmogorov law of large numbers. Weak convergence, its properties, Lévy theorem, Slutsky lemma, central limit theorems (CLT), Lindeberg-Feller fundamental theorem, Lindeberg condition, Berry-Esseen theorem, Multi-dimensional limit theorems. Populations, natural extensions in sample spaces, The problem of point statistical estimation, parametric and nonparametric caase, optimality criteria, asymptotic normality, Glivenko-Cantelli lemma, Vapnik-Chervonenkis inequality. Kernel density estimates and its properties.

Syllabus of tutorials:

Study Objective:

Knowledge:

In frame of the advanced course in Probability and Statistics on measure-theoretic level, to provide students with the knowledge necessary for the following future subjects using probability and stochastic models. To give a deeper insight into the field.

Abilities:

Orientation in majority of standard and advanced notions of the probability theory and statistics and capabilities of theoretical and practical applications in actual probabilistic computation.

Study materials:

Key:

[1] Rényi A., Foundations of probability, Holden-Day Inc., San Francisco, 1970.

[2] Schervish M.J., Theory of Statistics, Springer, 1995.

Recommended:

[3] Shao J., Mathematical Statistics, Springer, 1999.

[4] Lehmann E.L., Point Estimation, Wiley, N.Y., 1984.

[5] Lehmann E.L., Testing Statistical Hypotheses, Springer, N.Y., 1986.

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• BS Matematické inženýrství - Matematické modelování (elective course)
• BS Matematické inženýrství - Matematická fyzika (elective course)
• BS Matematické inženýrství - Aplikované matematicko-stochastické metody (compulsory course of the specialization, elective course)
• BS Informatická fyzika (elective course)
• BS Aplikace softwarového inženýrství (elective course)
• BS Aplikovaná informatika (elective course)
• BS jaderné inženýrství B (elective course)
• BS Jaderné inženýrství C (elective course)
• BS Dozimetrie a aplikace ionizujícího záření (elective course)
• BS Experimentální jaderná a částicová fyzika (elective course)
• BS Inženýrství pevných látek (elective course)
• BS Diagnostika materiálů (elective course)
• BS Fyzika a technika termojaderné fúze (elective course)
• BS Fyzikální elektronika (elective course)
• BS Jaderná chemie (elective course)