# 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

Code | Subject | Credits | Winter | Summer | |

NMSA401 | Primary Seminar | 2 | 0/2 C | — | |

NMSA403 | Optimisation Theory | 5 | 2/2 C+Ex | — | |

NMSA405 | Probability Theory 2 | 5 | 2/2 C+Ex | — | |

NMSA407 | Linear Regression | 8 | 4/2 C+Ex | — | |

NMSA409 | Stochastic Processes 2 | 8 | 4/2 C+Ex | — | |

NSZZ023 | Diploma Thesis I | 6 | — | 0/4 C | |

NSZZ024 | Diploma Thesis II | 9 | 0/6 C | — | |

NSZZ025 | Diploma Thesis III | 15 | — | 0/10 C |

#### 7.2 Elective Courses

*Set 1*

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

Code | Subject | Credits | Winter | Summer | |

NMEK450 | Econometrics Seminar 1 | 2 | — | 0/2 C | |

NMEK551 | Econometric Project Seminar | 5 | 0/2 C | — | |

NMST450 | Statistical Seminar 1 | 2 | — | 0/2 C | |

NMST551 | Statistical Workshop | 5 | 0/2 C | — | |

NMTP450 | Seminar on Probability 1 | 2 | — | 0/2 C | |

NMTP551 | Seminar on Probability 2 | 5 | 0/2 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.

Code | Subject | Credits | Winter | Summer | |

NMEK432 | Econometrics | 8 | — | 4/2 C+Ex | |

NMEK436 | Computational Aspects of Optimisation | 5 | — | 2/2 C+Ex | |

NMEK531 | Mathematical Economics | 5 | 2/2 C+Ex | — | |

NMEK532 | Optimisation with Applications to Finance | 8 | — | 4/2 C+Ex | |

NMFM431 | Investment Analysis | 5 | 2/2 C+Ex | — | |

NMFM437 | Mathematics in Finance and Insurance | 6 | 4/0 Ex | — | |

NMFM531 | Financial Derivatives 1 | 3 | 2/0 Ex | — | |

NMFM532 | Financial Derivatives 2 | 3 | 2/0 Ex | — | |

NMFM535 | Stochastic Analysis in Financial Mathematics | 5 | — | 2/2 C+Ex | |

NMFM537 | Credit Risk in Banking | 3 | 2/0 Ex | — | |

NMFP436 | Data Science 2 | 5 | — | 2/2 C+Ex | |

NMST431 | Bayesian Methods | 5 | 2/2 C+Ex | — | |

NMST432 | Advanced Regression Models | 8 | — | 4/2 C+Ex | |

NMST434 | Modern Statistical Methods | 8 | — | 4/2 C+Ex | |

NMST436 | Experimental Design | 5 | 2/2 C+Ex | — | |

NMST438 | Survey Sampling | 5 | 2/2 C+Ex | — | |

NMST440 | Advanced aspects of the R environment | 4 | — | 0/2 C | |

NMST442 | Matrix Computations in Statistics | 5 | — | 2/2 C+Ex | |

NMST531 | Censored Data Analysis | 5 | 2/2 C+Ex | — | |

NMST532 | Design and Analysis of Medical Studies | 5 | — | 2/2 C+Ex | |

NMST533 | Asymptotic Inference Methods | 3 | 2/0 Ex | — | |

NMST535 | Simulation Methods | 5 | — | 2/2 C+Ex | |

NMST537 | Time Series | 8 | 4/2 C+Ex | — | |

NMST539 | Multivariate Analysis | 5 | — | 2/2 C+Ex | |

NMST541 | Statistical Quality Control | 5 | — | 2/2 C+Ex | |

NMST543 | Spatial Statistics | 5 | 2/2 C+Ex | — | |

NMST552 | Statistical Consultations | 2 | — | 0/2 C | |

NMTP432 | Stochastic Analysis | 8 | — | 4/2 C+Ex | |

NMTP434 | Invariance Principles | 6 | — | 4/0 Ex | |

NMTP436 | Continuous Martingales and Counting Processes | 3 | — | 2/0 Ex | |

NMTP438 | Spatial Modelling | 8 | — | 4/2 C+Ex | |

NMTP532 | Ergodic Theory | 4 | — | 3/0 Ex | |

NMTP533 | Applied Stochastic Analysis | 5 | 2/2 C+Ex | — | |

NMTP535 | Selected Topics on Measure Theory | 3 | 2/0 Ex | — | |

NMTP537 | Limit Theorems for Sums of Random Variables | 3 | 2/0 Ex | — | |

NMTP539 | Markov Chain Monte Carlo Methods | 5 | 2/2 C+Ex | — | |

NMTP541 | Stochastic Geometry | 3 | — | 2/0 Ex | |

NMTP543 | Stochastic Differential Equations | 6 | 4/0 Ex | — | |

NMTP545 | Theory of Probability Distributions | 3 | 2/0 Ex | — |

#### 7.3 Recommended Optional Courses

Code | Subject | Credits | Winter | Summer | |

NMFM461 | Demography | 3 | — | 2/0 Ex | |

NMST570 | Selected topics in psychometrics | 3 | 1/1 C+Ex | — | |

NMST571 | Seminar in psychometrics | 2 | — | 0/2 C | |

NMTP561 | Malliavin calculus | 3 | 2/0 Ex | — | |

NMTP562 | Markov Processes | 6 | — | 4/0 Ex | |

NMTP563 | Selected Probability Topics for Statistics | 5 | — | 2/2 C+Ex | |

NMTP567 | Selected Topics on Stochastic Analysis | 3 | 2/0 Ex | — | |

NMTP570 | Heavy-Tailed Distributions | 3 | — | 2/0 Ex | |

NMTP576 | Conditional Independence Structures | 3 | — | 2/0 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*

Code | Subject | Credits | Winter | Summer | |

NMSA407 | Linear Regression | 8 | 4/2 C+Ex | — | |

NMSA409 | Stochastic Processes 2 | 8 | 4/2 C+Ex | — | |

NMSA403 | Optimisation Theory | 5 | 2/2 C+Ex | — | |

NMSA405 | Probability Theory 2 | 5 | 2/2 C+Ex | — | |

NMSA401 | Primary Seminar | 2 | 0/2 C | — | |

Optional and Elective Courses | 32 |

*2nd year*

Code | Subject | Credits | Winter | Summer | |

NSZZ023 | Diploma Thesis I | 6 | 0/4 C | — | |

NSZZ024 | Diploma Thesis II | 9 | — | 0/6 C | |

NSZZ025 | Diploma Thesis III | 15 | — | 0/10 C | |

Optional and Elective Courses | 30 |