Probability Theory

Module MA2409

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2011

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

available module versions
WS 2011/2SS 2011

Basic Information

MA2409 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

  • Catalogue of non-physics elective courses
Total workloadContact hoursCredits (ECTS)
270 h 90 h 9 CP

Content, Learning Outcome and Preconditions


Independence of sigma-algebras and random variables, existence of sequences of random variables, Kolmogorov's extension theorem, Borel-Cantelli lemmas, Kolmogorov's 0-1-law, weak and strong law of large numbers, characteristic functions, weak convergence, central limit theorem for L²-random variables, Lindeberg-Feller. Conditional expectations. Martingales: inequalities, convergence theorems, optional stopping theorem. Brownian motion.

Learning Outcome

After successful completion of the module students are able to understand and apply measure-theoretic probability theory, in particular, - the theory of sequences of i.i.d. random variables, - martingale theory, - the basic theory of Brownian motion.


MA1001 Analysis 1, MA1002 Analysis 2, MA2003 Measure and Integration, MA1401 Introduction to Probability Theory

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VO 4 Probability Theory Rolles, S. Dienstag, 08:15–09:45
Freitag, 08:30–10:00
sowie einzelne oder verschobene Termine
UE 2 Exercises to Probability Theory Conache, D. Rolles, S. Termine in Gruppen

Learning and Teaching Methods

lecture, exercise course, self-study assignments


course reserve, blackboard, exercise sheets


Rick Durrett: Probability: Theory and Examples, Duxbury advanced series, third edition, 2005. Achim Klenke: Probability Theory: A Comprehensive Course, Springer, 2008.

Module Exam

Description of exams and course work

The module examination is based on a written exam (60-90 minutes). Students have to know the basics of measure theoretical probability theory and can give adequate solutions to application problems in limited time. They have to deal with martingals and basics of Brownian motion.

Exam Repetition

There is a possibility to take the exam at the end of the semester.

Current exam dates

Currently TUMonline lists the following exam dates. In addition to the general information above please refer to the current information given during the course.

Fr, 30.9.2016, 15:30 bis 17:00 MW: 1801
bis 19.9.2016 (Abmeldung bis 23.9.2016)

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Nuclei, Particles, Astrophysics

A journey of discovery to understanding our world at the subatomic scale, from the nuclei inside atoms down to the most elementary building blocks of matter. Are you ready for the adventure?


Biological systems, from proteins to living cells and organisms, obey physical principles. Our research groups in biophysics shape one of Germany's largest scientific clusters in this area.