MA750 Probability theory

Faculty of Science
Autumn 2018
Extent and Intensity
2/1. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
Mgr. Ondřej Pokora, Ph.D. (lecturer)
Guaranteed by
doc. PaedDr. RNDr. Stanislav Katina, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 17. 9. to Fri 14. 12. Mon 10:00–11:50 M6,01011
  • Timetable of Seminar Groups:
MA750/01: Mon 17. 9. to Fri 14. 12. Mon 12:00–12:50 M6,01011, O. Pokora
Prerequisites
Theoretical knowledge and practise in the scope of undergraduate courses of probability, mathematical statistics and calculus.
The aim of this course is to extend the knowledge of the probability theory. Principles known from the undergraduate courses of probability and mathematical statistics will be generalized, selected topics form the measure theory will be added and shown both on the theoretical as well as practical examples.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
This course is designed mainly as the extension of the knowledge of probability theory. The lectures go back to the basic course of the probability and statistics and add new terms and detailed explanations. Although the content is primarily in the formal level of the mathematical theory, each topic is motivated by some well-known problem (e. g., picking a random number from an interval, infinite coin tossing), which is intuitive for the student. The new terms and features are then demonstrated in these examples.
Learning outcomes
After completing this course, the student will be able to:
- construct the probability space, the Lebesgue and counting measure, random variables and probability distributions in complicated problems;
- calculate the Lebesque-Stieltjes integral and the expectations with respect to the measure;
- define and use the moment generating, cumulative distribution and characteristic function;
- understand the convergences of random variables and to apply the central limit theorem;
- use the conditional probability, expectation, filtration, and martingales in complicated problems.
Syllabus
  • Probability, probability space.
  • Measure, measurable functions, random variables, Lebesgue-Stieltjes measure, Lebesgue-Stieltjes integral.
  • Independence, continuity, expected value.
  • Cumulative distribution function, characteristic function, moment generating function.
  • Central limit theorem, law of large numbers.
  • Distributions of random variables.
  • Radon-Nikodym theorem, conditional probability, conditional expectation.
  • Martingales, stopping time.
  • Applications.
Literature
  • ROSENTHAL, Jeffrey S. A first look at rigorous probability theory. 2nd ed. Hackensack, N.J.: World Scientific, 2006, xvi, 219. ISBN 9789812703705. info
  • RIEČAN, Beloslav. Miniteória pravdepodobnosti. Banská Bystrica: Vydavateľstvo Belianum, Univerzita Mateja Bela, 2015, 52 pp. URL info
  • O pravdepodobnosti a miere. Edited by Beloslav Riečan. 1. vyd. Bratislava: Alfa, vydavateľstvo technickej a ekonomickej literatúry, 1972, 157 s. info
  • LACHOUT, Petr. Teorie pravděpodobnosti. 2. vyd. Praha: Karolinum, 2004, 146 s. ISBN 8024608723. info
  • ROUSSAS, George G. An Introduction to Measure-Theoretic Probability. 2014, 426 pp. ISBN 978-0-12-800042-7. info
  • BILLINGSLEY, Patrick. Probability and measure. 3rd ed. New York: Wiley, 1995, xii, 593. ISBN 0471007102. URL info
Teaching methods
Lectures: 2 hours a week. Practical classes: 1 hour a week.
Assessment methods
Practical classes: active participation, homeworks. Final exam: written and oral part, at least 50 % of points in each part required to pass.
Language of instruction
Czech
Further Comments
Study Materials
The course is taught annually.
The course is also listed under the following terms autumn 2017, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Autumn 2018, recent)
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