## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-60-11-M-4

## Module Identification

Module Number Module Name CP (Effort)
MAT-60-11-M-4 Probability Theory 9.0 CP (270 h)

## Basedata

CP, Effort 9.0 CP = 270 h 1 Sem. in WiSe [4] Bachelor (Specialization) [EN] English Ritter, Klaus, Prof. Dr. (PROF | DEPT: MAT) Sass, Jörn, Prof. Dr. (PROF | DEPT: MAT) Grothaus, Martin, Prof. Dr. (PROF | DEPT: MAT) Korn, Ralf, Prof. Dr. (PROF | DEPT: MAT) Ritter, Klaus, Prof. Dr. (PROF | DEPT: MAT) Sass, Jörn, Prof. Dr. (PROF | DEPT: MAT) [MAT-STO] Stochastics/Statistics/Financial Mathematics [MAT-88.105-SG] M.Sc. Mathematics [NORM] Active

## Notice

Without a proof of successful participation in the exercise classes, only 6 credit points will be awarded for the module.

## Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-60-11-K-4
Probability Theory
P 84 h 186 h
UK-Schein
- PL1 9.0 WiSe
• About [MAT-60-11-K-4]: Title: "Probability Theory"; Presence-Time: 84 h; Self-Study: 186 h
• About [MAT-60-11-K-4]: The study achievement "[UK-Schein] proof of successful participation in the exercise classes (incl. written examination, ungraded)" must be obtained.

## Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: each semester
• Examination number: 84270 ("Probability Theory")

## Contents

• Types of convergence (stochastic, almost sure, weak, L_p-convergence, convergence in distribution),
• Characteristic functions,
• Sums of independent random variables,
• Strong laws of large numbers, variants of the central limit theorem,
• Conditional expectation,
• Martingales in discrete time,
• Brownian motion.

## Competencies / intended learning achievements

Upon completion of the module, the students have gained advanced knowledge in stochastics and the foundations required for further research in the field of stochastic processes. They understand the proofs presented in the lecture and are able to comprehend and explain them.

By completing exercises, the students will have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work.

## Literature

• H. Bauer: Probability Theory,
• P. Billingsley: Probability and Measure,
• P. Gänssler, W. Stute: Wahrscheinlichkeitstheorie,
• A. Klenke: Wahrscheinlichkeitstheorie.

## Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de)

None

## References to Module / Module Number [MAT-60-11-M-4]

Course of Study Section Choice/Obligation
[MAT-88.105-SG] M.Sc. Mathematics [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
[MAT-88.105-SG] M.Sc. Mathematics [Core Modules (non specialised)] Applied Mathematics [WP] Compulsory Elective
[MAT-88.118-SG] M.Sc. Industrial Mathematics [Core Modules (non specialised)] General Mathematics [WP] Compulsory Elective
[MAT-88.276-SG] M.Sc. Business Mathematics [Core Modules (non specialised)] General Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International [Core Modules (non specialised)] Pure Mathematics [WP] Compulsory Elective
[MAT-88.706-SG] M.Sc. Mathematics International [Core Modules (non specialised)] Applied Mathematics [WP] Compulsory Elective