German original

# Probability Theory (dt. Wahrscheinlichkeitstheorie)

 Level, degree of commitment in original study programme Advanced module, compulsory elective module Forms of teaching and learning,workload Lecture (4 SWS), recitation class (2 SWS), 270 hours (90 h attendance, 180 h private study) Credit points,formal requirements 9 CP Course requirement: Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination Language,Grading German,The grading is done with 0 to 15 points according to the examination regulations for study course M.Sc. Business Mathematics. Original study programme M.Sc. Wirtschaftsmathematik / Mathematische Vertiefungs- und Praxismodule Duration,frequency One semester, each winter semester Person in charge of the module's outline Prof. Dr. Hajo Holzmann

## Contents

The basic concepts of probability theory, based on measure and integration theory, are discussed, in particular

• General probability spaces, random variables
• Independence, laws of large numbers
• weak convergence, characteristic functions and central limit theorem
• conditional expectations, conditional distributions, martingales
• stochastic processes, in particular Brownian motion

## Qualification Goals

The students shall

• learn the basics of probability theory in a mathematically rigorous way, based on measure theory,
• practice mathematical methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof methods),
• improve their oral communication skills in the recitation classes by practicing free speech in front of an audience and during discussion.

## Prerequisites

None. The competences taught in the following modules are recommended: either Foundations of Mathematics and Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Measure and Integration Theory, either Elementary Stochastics [Bachelor Module] or Elementary Stochastics [Lehramt Module].

• Bauer, H., „Wahrscheinlichkeitstheorie“, de Gruyter 2004.
• Billingsley, P., „Probability and Measure“, John Wiley & Sons 1995
• Durrett, R., „Probability Theory and Examples“, Wadsworth & Brooks 1991
• Klenke, A., „Wahrscheinlichkeitstheorie“, Springer 2008