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

7.2 Elective Courses

Set 1

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

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.

7.3 Recommended Optional Courses

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

2nd year