Course detail

Mathematical Foundations of Risk Analysis

ÚSI-RSMATAcad. year: 2025/2026

The course concerns mathematical modelling and its applications in risk engineering. It explains fundamental ideas and notions, especially by means of suitable examples, and deals with their applicability and a unifying view of mathematical principles. Related mathematical methods for achieving solutions in individual areas will be presented with the use of suitable software: Statistics, Minitab, Matlab and Excel.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Zdeněk Karpíšek, CSc.

Department

Institute of Forensic Engineering (ÚSI)

Entry knowledge

Basic knowledge of undergraduate mathematics (linear algebra, differential and integral calculus, probability and statistics, numerical methods), and computer technology for use with application software.

Rules for evaluation and completion of the course

Course unit credit requirements: active participation in seminars, mastery of the subject matter, and the submission of a semestral assignment. Examination (written form): a practical part (5 tasks), a theoretical part (5 tasks); ECTS evaluation used.
Attendance at seminars is monitored, and the teacher decides on the manner of compensation for absences.

Aims

Students will gain useful knowledge of mathematical models used in risk modelling. They will also learn how to apply the studied models and methods to technical phenomena and processes.
Fundamental concepts, methods and analytical techniques related to risk modelling will be studied. Specific ways of reasoning that are typical for risk analysis and related model building will be developed and enhanced.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

ANDĚL, Jiří. Základy matematické statistiky. Praha: Matfyzpress, 2011. ISBN 978-80-7378-162-0. (CS)
CIPRA, Tomáš. Riziko ve financích a pojišťovnictví: Basel III a Solvency II. Ekopress 2015. ISBN 978-80-87865-24-8.
KARPÍŠEK, Zdeněk. MATEMATIKA IV: Statistika a pravděpodobnost. Akademické nakladatelství CERM s.r.o., Brno 2014. ISBN 978-80-214-4858-2.
MONTGOMERY, Douglas C., RUNGER, George. Applied Statistics and Probability for Engineers. 5th ed. New York: John Wiley & Sons, 2010. ISBN 978-0-470-05304-1. (CS)

Recommended reading

AGRESTI, Alan. Categorical Data Analysis. 3rd ed. New York: John Wiley & Sons, 2013. ISBN 0-470-46363-5.
BROCKWELL, Peter J., DAVIS, Richard, A. Introduction to Time Series and Forecasting. 2nd ed. New York: Springer-Verlag, 2002. ISBN 0-387-95351-5
KLIR, George J., YUAN, Bo. Fuzzy Sets and Fuzzy Logic - Theory and Applications. New Jersey: Prentice Hall, 1995.

Classification of course in study plans

  • Programme RRTES_P Master's

    specialization RRTS , 1 year of study, winter semester, compulsory
    specialization RRES , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Karpíšek, CSc.

Syllabus

1. Fundamental mathematical concepts of risk engineering.
2. Selected deterministic models for economic and financial computations.
3. Selected deterministic models for numerical and engineering approaches, sensitivity analysis.
4. Uncertainty in risk engineering problems - stochastic and fuzzy models.
5. Problems of system reliability and risk evaluation modelling, simulation approaches.
6. Elementary models of decision making under parametric uncertainty.
7. Selected methods of estimating the probability distributions of model parameters. Statistical software.
8. Advanced mathematical statistics methods - linear and nonlinear multiple regression analysis.
9. Elements of categorical, factor and cluster analysis.
10. Parametric and nonparametric statistical hypothesis tests.
11. Models for dynamic problems - introduction to Markov chains (applications in production systems).
12. Elements of time series analysis.
13. Basic models for the quality control of production processes and products.

Exercise

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Zdeněk Karpíšek, CSc.

Syllabus

1. Fundamental mathematical concepts of risk engineering.
2. Selected deterministic models for economic and financ computations.
3. Selected deterministic models for numerical and engineering approaches, sensitivity analysis.
4. Uncertainty in problems of risk engineering - stochastic and fuzzy models.
5. Problems of systems reliability and risks evaluations modeling, simulation approach.
6. Elementary models of decision making under uncertainty.
7. Selected estimation methods of models parameters probability distributions. Statistical software.
8. Advanced mathematical statistics methods - linear and nonlinear multidimensional regression analysis.
9. Elements of categorical, factor and cluster analysis.
10. Parametric and nonparametric statistical hypotheses tests.
11. Models for dynamic problems - introduction to Markov chains (applications in production systems).
12. Elements of time series analysis.
13. Basic models for quality control of production and produces.