Course detail

Mathematics 4

FAST-BAA004Acad. year: 2025/2026

Discrete and continuous random variable and vector, probability function, density function, probability, cumulative distribution, transformation of random variables, independence of random variables, numeric characteristics of random variables and vectors, special distribution laws.
Random sample, point estimation of an unknown distribution parameter and its properties, interval estimation of a distribution parameter, testing of statistical hypotheses, tests of distribution parameters, goodness-of-fit tests, basics of regression analysis.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

Ing. Jan Holešovský, Ph.D.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Offered to foreign students

Of all faculties

Entry knowledge

Basic knowledge of the theory of one and more functions (derivative, partial derivative, limit and continuous functions, graphs of functions). Ability to calculate definite integrals, double and triple integrals and knowledge of their basic applications.

Rules for evaluation and completion of the course

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Aims

The students should get an overview of the basic properties of probability to be able to deal with simple practical problems dealing with stochastic uncertainty. They should get familiar with the basic statistical methods used for point and interval estimates, testing statistical hypotheses, and linear model. Student will be able to solve simple practical probability problems and to use basic statistical methods, estimates of parameters and parametric functions, testing statistical hypotheses, and linear models.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

DEVORE, J. L.; BERK, K. N. and CARLTON, M. A. Modern mathematical statistics with applications. Third edition. Cham: Springer, 2021. ISBN 978-3-030-55158-2. (EN)
KOUTKOVÁ, H., DLOUHY, O. Sbírka příkladů z pravděpodobnosti a matematické statistiky. Brno: CERM, 2011. 63 s. ISBN 978-80-7204-740-6.  (CS)
KOUTKOVÁ, H. Elektronické studijní opory. M03 - Základy teroie odhadu, M04 - Základy testování hypotéz. FAST VUT Brno, 2004. [https://intranet.fce.vutbr.cz/pedagog/predmety/opory.asp ]  (CS)
KOUTKOVÁ, H. Základy teorie odhadu. Brno: CERM, 2007. 51 s. ISBN 978-80-7204-527-3.   (CS)
KOUTKOVÁ, H. Základy testování hypotéz. Brno: CERM, 2007. 52 s. ISBN 978-80-7204-528-0.  (CS)
KOUTKOVÁ, H., MOLL, I. Základy pravděpodobnosti. Brno: CERM, 2011.127 s. ISBN 978-80-7204-738-3.   (CS)
KAPTEIN, M. and HEUVEL van den, E. Statistics for data scientists: an introduction to probability, statistics, and data analysis. Cham: Springer, 2022. ISBN 9783030105303. (EN)
NEUBAUER, J., SEDLAČÍK, M. a KŘÍŽ, O. Základy statistiky: Aplikace v technických a ekonomických oborech - 3., rozšířené vydání. Grada, 2021. ISBN 978-80-271-4484-6.  (CS)

Recommended reading

MATHAI, A. M. and HAUBOLD, H. J. Probability and Statistics: A Course for Physicists and Engineers. Berlin/Boston: De Gruyter, 2017. ISBN 9783110562545.  (EN)
RAMACHANDRAN, K.M. and TSOKOS, C. P. Mathematical Statistics with Applications in R. 3rd edition. San Diego: Elsevier Science & Technology, 2020. ISBN 9780128178157.  (EN)
WALPOLE, R.E., MYERS, R.H. Probability and Statistics for Engineers and Scientists. 8th ed. London: Prentice Hall, Pearson education LTD, 2007. 823 p. ISBN 0-13-204767-5.   (EN)

Classification of course in study plans

  • Programme BPA-SI Bachelor's 3 year of study, winter semester, compulsory

  • Programme BPC-SI Bachelor's

    specialization S , 3 year of study, winter semester, compulsory
    specialization K , 3 year of study, winter semester, compulsory
    specialization E , 3 year of study, winter semester, compulsory
    specialization M , 3 year of study, winter semester, compulsory
    specialization V , 3 year of study, winter semester, compulsory

  • Programme BPC-MI Bachelor's 2 year of study, winter semester, compulsory
  • Programme BPC-EVB Bachelor's 3 year of study, winter semester, compulsory
  • Programme BKC-SI Bachelor's 3 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Karel Mikulášek, Ph.D.
RNDr. Veronika Chrastinová, Ph.D.
RNDr. Mgr. Lucie Zrůstová, Ph.D.
Ing. Jan Holešovský, Ph.D.
Mgr. Darina Brothánková, Ph.D.

Syllabus

  1. Random events (basic properties, operations), probability (classical, axiomatic) and its properties.
  2. Conditional probability and the law of total probability, Bayes' theorem, independence of random events.
  3. Random variable: introduction, cumulative distribution function, density function and probability mass function.
  4. Numeric characteristics of random variables: mean, variance, standard deviation, modus, quantiles. Rules of calculation mean and variance.
  5. Discrete probability distributions: Bernoulli, binomial, hypergeometric and Poisson.
  6. Continuous probability distributions: uniform, normal, chi2, Student's and Fisher-Snedecor distribution.
  7. Bivariate discrete random vector, joint and marginal distributions, independence of the components, numeric characteristics.
  8. Random sample and sample statistics (properties, their distribution for sample from N). Central limit theorem.
  9. Point estimates (unbiased, best, consistent) and interval estimates for parameters of normal and Bernoulli random variables.
  10. Testing statistical hypothesis: principle and one-sample tests (z test, t test, chi2 test for variance, asymptotic test for the parameter of Bernoulli distribution).
  11. Two-sample tests: F test, t test for unknown variances under homoscedasticity or heteroscedasticity, paired t test, equality test for parameters of two Bernoulli distributions.
  12. Goodness-of-fit tests: chi2 test, graphical diagnostics (histogram, QQ plot, PP plot), and some alternatives.
  13. Introduction to regression analysis.

Exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Karel Mikulášek, Ph.D.
Ing. Pavel Špaček, Dr.
RNDr. Veronika Chrastinová, Ph.D.
RNDr. Mgr. Lucie Zrůstová, Ph.D.
Ing. Jan Holešovský, Ph.D.
Mgr. Darina Brothánková, Ph.D.
Ing. Matouš Cabalka

Syllabus

  1. Descriptive statitics for univariate data.
  2. Classical probability and its calculation, application of basec properties.
  3. Conditional probability and the law of total probabilty, Bayes' rule and independence of random events.
  4. Functional and numerical characteristis of random variables.
  5. Functional and numerical characteristis of random variables - continuation.
  6. Transformation of random variable. Discrete probability distributions.
  7. Discrete (binomial, hypergeometric, Poisson) and continuous (normal) probability distributions.
  8. Test. Approximation of distributions.
  9. Bivariate discrete random vector: functional and numerical characteristics, independence of its components.
  10. Point and interval estimates for parameters of normal and Bernoulli random variables.
  11. One-sample tests of hypotheses about the parameters of normal and Bernoulli distributions.
  12. Two-sample tests of hypotheses about the parameters of normal and Bernoulli distributions.
  13. Goodness-of-fit tests.