Course Details

Mathematics 4

Academic Year 2025/26

BAA004 course is part of 9 study plans

​ BPA-SI Winter Semester 3rd year

​ BPC-SI / S Winter Semester 3rd year

​ BPC-SI / K Winter Semester 3rd year

​ BPC-SI / E Winter Semester 3rd year

​ BPC-SI / M Winter Semester 3rd year

​ BPC-SI / V Winter Semester 3rd year

​ BPC-MI Winter Semester 2nd year

​ BPC-EVB Winter Semester 3rd year

​ BKC-SI Winter Semester 3rd year

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.

Credits

5 credits

Language of instruction

Czech, English

Semester

winter

Course Guarantor

Ing. Jan Holešovský, Ph.D.

Institute

Institute of Mathematics and Descriptive Geometry

Forms and criteria of assessment

course-unit credit and examination

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.

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.

Basic Literature

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)
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)
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)
KOUTKOVÁ, H., MOLL, I. Základy pravděpodobnosti. Brno: CERM, 2011.127 s. ISBN 978-80-7204-738-3.   (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., DLOUHY, O. Sbírka příkladů z pravděpodobnosti a matematické statistiky. Brno: CERM, 2011. 63 s. ISBN 978-80-7204-740-6.  (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)

Offered to foreign students

To offer to students of all faculties

Course on BUT site

https://www.vut.cz/en/students/courses/detail/289590

Lecture

13 weeks, 2 hours/week, elective

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

13 weeks, 2 hours/week, compulsory

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.