Selected topics from probability theory for physicists

Garant předmětu:

Michal Šumbera

Lecturer:

Michal Šumbera

Tutor:

Michal Šumbera

Supervisor:

Department of Physics

Synopsis:

Discrete and continuous probability distributions (Binomial, Poisson, negative binomial, normal, etc.) as well as the processes that lead to their origin have long played a major role in physics, biology and economics. The impetus for the further expansion of these divisions in the 20th century was their application to the description of neutron cascades, multiple particle production and the spread of infectious diseases. The generalization of the properties of these distributions has later on led to the discovery of new classes of distributions - infinitely divisible and stable distributions, which are currently widely used in physics and finance.

Requirements:

credit test.

Syllabus of lectures:

1. Discrete probability distributions and their characteristics. Generating functions. Markov processes.

2. Compound probability distributions. Bienaymé-Galton-Watson process.

3. Branching processes and their applications in neutron transport, production of cosmic ray showers and in biology.

4. Infinitely divisible distributions. Combinants and cumulants. Their use in statistical mechanics.

5. Sibuya and other discrete distributions connected with it. Composite models.

6. Thinning of discrete distributions. Self-decomposable and stable distributions.

7. Continuous stable distributions. Cauchy, Landau and Lévy distributions. Lévy process.

8. Renormalization (semi) group.

Syllabus of tutorials:

Study Objective:

Study materials:

Recommended literature:

[1] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, John Wiley and Sons, Inc., New York 1968.

[2] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley and Sons, Inc., New York 1971.

[3] Norman L. Johnson, Adrienne W. Kemp, Samuel Kotz, Univariate Discrete Distributions, John Wiley and Sons, 2005.

[4] Theodore E. Harris, The Theory of Branching Processes, The Rand Corporation, Santa Monica, California, 1964.

[5] F.W. Steutel, K. van Harn, Infinite divisibility of probability distributions on the real line, Marcel Dekker, New York, 2004.

[6] V.I. Uchaikin, V.M. Zolotarev, Chance and Stability, Stable Distributions and their Applications, De Gruyter 1999, ISBN-10: 9067643017.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Jaderná a částicová fyzika (elective course)