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# 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 2020

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

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2021/2SS 2020SS 2019WS 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
• Specialization Modules in Elite-Master Program Theoretical and Mathematical Physics (TMP)
270 h 90 h 9 CP

### Content, Learning Outcome and Preconditions

#### Content

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.

#### 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, in particular laws of large numbers and the central limit theorem
- martingale theory.

#### Preconditions

MA1001 Analysis 1, MA1002 Analysis 2, MA2003 Measure and Integration, MA1401 Introduction to Probability Theory
Bachelor 2019: MA0001 Analysis 1, MA0002 Analysis 2, MA0003 Analysis 3, MA0009 Introduction to Probability Theory and Statistics

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 4 Probability Theory [MA2409] Berger Steiger, N. Conache, D. Tue, 08:30–10:00, GALILEO 300
Fri, 08:30–10:00, MI HS1
Tue, 08:30–10:00, MI HS2
and singular or moved dates
UE 2 Exercises to Probability Theory [MA2409] Berger Steiger, N. Conache, D. dates in groups

#### Learning and Teaching Methods

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

#### Media

course reserve, blackboard, exercise sheets

#### Literature

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 (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.

#### Exam Repetition

The exam may be repeated 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.

Title
TimeLocationInfoRegistration
Probability Theory
Fri, 2023-02-17, 8:30 till 10:00 till 2023-01-15 (cancelation of registration till 2023-02-10)
Wed, 2023-04-12, 8:00 till 9:30 till 2023-03-27 (cancelation of registration till 2023-04-05)
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