Probability, Mathematical Statistics and Econometrics

Coordinated by: Department of Probability and Mathematical Statistics
Study branch coordinator: doc. Ing. Marek Omelka, Ph.D.

The curriculum is targeted at students who wish to obtain theoretical and practical knowledge about the mathematics of random events. It is primarily characterized by a balance between rigorous mathematical theory, depth of insight into various fields of the subject (probability, statistics, econometrics), and applications in various areas. The students will obtain a general background by taking compulsory courses in probability, optimization, linear regression and random processes. They continue by taking elective courses to extend their expertise and choose a specialization they wish to study more deeply. In seminars, they learn to work independently as well as to collaborate on complex projects. Great emphasis is placed on the development of independent analytical thinking. Probability, statistics and econometrics have a close relationship to other mathematical subjects (mathematical analysis, numerical mathematics, discrete mathematics). Applications are inspired by real problems from economics, medicine, technology, natural sciences, physics and computer science. The primary objective of the programme is to prepare students for successful careers in academia as well as in finance, telecommunications, marketing, medicine and natural sciences.

The graduate will be familiar with mathematical modelling of random events and processes and its applications. He/she will understand the foundations of probability theory, mathematical statistics, random process theory and optimization. His/her general background will have been extended to a deeper knowledge of random process theory and stochastic analysis, modern statistical methods, or advanced optimization and time series analysis. The graduate will understand the substance of the methods, grasp their mutual relationships, and will be able to actively extend them and use them. He/she will know how to apply theoretical knowledge to practical problems in a creative way. The graduate's ability to think logically, to analyse problems, and to solve non-trivial problems can be put to use in independent and creative jobs in the commercial sector or in academic positions.

Assumed knowledge

It is assumed that an incoming student of this branch has sufficient knowledge of the following topics and fields:

Differential and integral calculus of one variable and several variables. Measure theory. Lebesgue integral. Vector spaces, matrix algebra. Foundations of complex and functional analysis.
Foundations of probability theory.
Foundations of mathematical statistics and data analysis.
Markov chain theory.

Should an incoming student not meet these entry requirements, the coordinator of the study programme may assign a method of acquiring the necessary knowledge and abilities, which may for example mean taking selected bachelor's courses, taking a reading course with an instructor, or following tutored independent study.

7.1 Obligatory Courses

NMSA401Primary Seminar 20/2 C
NMSA403Optimisation Theory 52/2 C+Ex
NMSA405Probability Theory 2 52/2 C+Ex
NMSA407Linear Regression 84/2 C+Ex
NMSA409Stochastic Processes 2 84/2 C+Ex
NSZZ023Diploma Thesis I 60/4 C
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 C

7.2 Elective Courses

Set 1

It is required to earn at least 7 credits from this group.

NMEK450Econometrics Seminar 1 20/2 C
NMEK551Econometric Project Seminar 50/2 C
NMST450Statistical Seminar 1 20/2 C
NMST551Statistical Workshop 50/2 C
NMTP450Seminar on Probability 1 20/2 C
NMTP551Seminar on Probability 2 50/2 C

Set 2

It is required to earn at least 43 credits from this group. We recommend making a planned choice of subject areas for the final exam and the master's thesis topic when choosing elective courses.

NMEK432Econometrics 84/2 C+Ex
NMEK436Computational Aspects of Optimisation 52/2 C+Ex
NMEK531Mathematical Economics 52/2 C+Ex
NMEK532Optimisation with Applications to Finance 84/2 C+Ex
NMFM431Investment Analysis 52/2 C+Ex
NMFM437Mathematics in Finance and Insurance 64/0 Ex
NMFM531Financial Derivatives 1 32/0 Ex
NMFM532Financial Derivatives 2 32/0 Ex
NMFM535Stochastic Analysis in Financial Mathematics 52/2 C+Ex
NMFM537Credit Risk in Banking 32/0 Ex
NMFP436Data Science 2 52/2 C+Ex
NMST431Bayesian Methods 52/2 C+Ex
NMST432Advanced Regression Models 84/2 C+Ex
NMST434Modern Statistical Methods 84/2 C+Ex
NMST436Experimental Design 52/2 C+Ex
NMST438Survey Sampling 52/2 C+Ex
NMST440Advanced aspects of the R environment 40/2 C
NMST442Matrix Computations in Statistics 52/2 C+Ex
NMST531Censored Data Analysis 52/2 C+Ex
NMST532Design and Analysis of Medical Studies 52/2 C+Ex
NMST533Asymptotic Inference Methods 32/0 Ex
NMST535Simulation Methods 52/2 C+Ex
NMST537Time Series 84/2 C+Ex
NMST539Multivariate Analysis 52/2 C+Ex
NMST541Statistical Quality Control 52/2 C+Ex
NMST543Spatial Statistics 52/2 C+Ex
NMST552Statistical Consultations 20/2 C
NMTP432Stochastic Analysis 84/2 C+Ex
NMTP434Invariance Principles 64/0 Ex
NMTP436Continuous Martingales and Counting Processes 32/0 Ex
NMTP438Spatial Modelling 84/2 C+Ex
NMTP532Ergodic Theory 43/0 Ex
NMTP533Applied Stochastic Analysis 52/2 C+Ex
NMTP535Selected Topics on Measure Theory 32/0 Ex
NMTP537Limit Theorems for Sums of Random Variables 32/0 Ex
NMTP539Markov Chain Monte Carlo Methods 52/2 C+Ex
NMTP541Stochastic Geometry 32/0 Ex
NMTP543Stochastic Differential Equations 64/0 Ex
NMTP545Theory of Probability Distributions 32/0 Ex

7.3 Recommended Optional Courses

NMFM461Demography 32/0 Ex
NMST570Selected topics in psychometrics 31/1 C+Ex
NMST571Seminar in psychometrics 20/2 C
NMTP561Malliavin calculus 32/0 Ex
NMTP562Markov Processes 64/0 Ex
NMTP563Selected Probability Topics for Statistics 52/2 C+Ex
NMTP567Selected Topics on Stochastic Analysis 32/0 Ex
NMTP570Heavy-Tailed Distributions 32/0 Ex
NMTP576Conditional Independence Structures 32/0 Ex

7.4 State Final Exam

Requirements for taking the final exam

Earning at least 120 credits during the course of the study.
Completion of all obligatory courses prescribed by the study plan.
Earning at least 7 credits by completion of elective courses from group I.
Earning at least 43 credits by completion of elective courses from group II.
Submission of a completed master's thesis by the submission deadline.

Oral part of the state final exam

The oral part of the final exam consists of three subject areas. The first subject area is common. The second subject area is selected from three options (2A, 2B, 2C). The third subject area is selected from seven options 3A–3G. One question is asked from the common subject area and one from each selected optional subject area.

Requirements for the oral part of the final exam

Common subject area

1. Foundations of Probability, Statistics and Random Processes
Foundations of Markov chain theory. Stationary sequences and processes. Linear regression model. Conditional expectation. Martingales in discrete time. Optimization, linear and non-linear programming.

Optional subject area 2: Advanced Models

A choice of one of three options

2A. Econometrics and Optimization Methods
Stationary sequences, time series. Foundations of econometrics. Advanced optimization.

2B. Advanced Statistical Analysis.
Modern theory of estimation and statistical inference. Regression models for non-normal and correlated data.

2C. Processes in Time and Space.
Stochastic processes in continuous time. Martingales. Invariance principles. Brownian motion.

Optional subject area 3: Special Topics

A choice of one of seven options

3A. Econometric Models
Mathematical economics. Time series with financial applications. Advanced econometrical and statistical models. Multivariate statistical analysis.

3B: Optimization Methods
General optimization problems, optimal control. Applications of optimization in economics and finance. Mathematical economics. Time series.

3C: Spatial Modelling
Spatial modelling and spatial statistics. Foundations of stochastic analysis. Limit theorems in probability theory.

3D: Stochastic Analysis
Stochastic analysis. Itô formula. Stochastic differential equations. Poisson processes, stationary point processes. Limit theorems.

3E. Statistics in Industry, Trade and Business
Survey sampling. Design of industrial experiments. Time series. Statistical quality control. Reliability theory.

3F. Statistics in Natural Sciences
Design and analysis of medical experiments. Multivariate statistical analysis. Survival analysis. Bayesian methods.

3G. Theoretical Statistics
Invariance principles. Limit theorems. Methods for censored data analysis. Multivariate analysis.

7.4. Recommended Course of Study

1st year

NMSA407Linear Regression 84/2 C+Ex
NMSA409Stochastic Processes 2 84/2 C+Ex
NMSA403Optimisation Theory 52/2 C+Ex
NMSA405Probability Theory 2 52/2 C+Ex
NMSA401Primary Seminar 20/2 C
 Optional and Elective Courses 32  

2nd year

NSZZ023Diploma Thesis I 60/4 C
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 C
 Optional and Elective Courses 30