MA750 Probability theory

Faculty of Science
Autumn 2024
Extent and Intensity
2/1. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
Mgr. Ondřej Pokora, Ph.D. (lecturer)
Guaranteed by
doc. Mgr. Jan Koláček, Ph.D.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Wed 16:00–17:50 M6,01011
  • Timetable of Seminar Groups:
MA750/01: Wed 18:00–18:50 M6,01011, O. Pokora
Prerequisites
Theoretical knowledge and practise in the scope of undergraduate courses of probability, mathematical statistics and calculus are required. Basic knowledge of discrete mathematics and calculus is also assumed.
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
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.
The course is designed mainly as the theoretical 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 theoretical 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 expectation and Lebesque-Stieltjes integral and integrals respect to a measure;
- understand different concepts of convergences of random variables;
- use Radon-Nikodym derivative for changing the probability measure, classify probability distributions and understand the concepts of probability density function;
- construct the conditional probability, expectation, filtration, and martingales in complicated problems and apply them.
Syllabus
  • Probability space and its construction, Extension theorem.
  • Measure, outer measure, measurable sets.
  • Continuity of probability, set limits.
  • Random variable and its construction, independence.
  • Expected value, integral with respect to measure.
  • Inequalities for random variables, convergence of random variables, laws of large numbers.
  • Probability distribution, cumulative distribution function, probability density funcion.
  • Weak convergence, moment generating function, characteristic function, Central limit theorem.
  • Radon-Nikodym theorem, Radon-Nikodym derivative, decomposition of probability distribution.
  • Conditional probability, conditional expectation.
  • Filtration, martingale, stopping time.
  • Examples.
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
Classes are in full-time form every week: joint block of a lecture (2 hours) and a seminar (1 hour).
Assessment methods
Active participation and discussions in lectures and seminars, solving interim problems. Final examination: full-time form – written and oral part. For successful completion, it is necessary to achieve at least 50% of the maximum achievable number of points in each part.
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
Teacher's information
https://is.muni.cz/auth/el/sci/podzim2024/MA750/index.qwarp
The lessons are usually in Czech or in English as needed, and the relevant terminology is always given with English equivalents. The target skills of the study include the ability to use the English language passively and actively in their own expertise and also in potential areas of application of mathematics. Assessment in all cases may be in Czech and English, at the student's choice.
The course is also listed under the following terms autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023.
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