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Lecturer(s)
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Bůžek Pavel, Ing. Mgr.
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Chmelík Slavomil, PhDr. Ph.D.
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Course content
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1. Basic terms from combinatorics. Elementary and random phenomena. Classical definition of probability and its generalization. Space of elementary phenomena, probability space. 2. Axiomatic introduction of probability space. Basic set, set of random phenomena and probability measure P. Properties of individual parts of the probability space. 3. Operations with random phenomena. Conditional probability. Independence of random phenomena. Bayes theorem. Geometric probability. 4. Probability models (selection with and without return, Maxwell-Boltzman scheme, and others) 5. Introduction of the term random variable. Probability function, distribution function. 6. Parameters of random variables. Mean value of random variables, variance, standard deviation. 7. Introduction and properties of the most important discrete random variables. 8. Introduction and properties of the most important continuous random variables. 9. Independent random variables. Covariance and correlation ratio. Methods of measuring the strength of the relationship between two or more random variables. 10. Random vectors - basic characteristics. Multivariate random variables. Marginal distribution function. 11. Law of large numbers. The concept of convergence on the space of random variables. Chebyshev's inequality. Chebyshev's theorem. Central Limit Theorem.
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Learning activities and teaching methods
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Lecture with practical applications, Individual study, Practicum
- Contact hours
- 52 hours per semester
- Undergraduate study programme term essay (20-40)
- 20 hours per semester
- Preparation for an examination (30-60)
- 35 hours per semester
- Preparation for formative assessments (2-20)
- 10 hours per semester
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| prerequisite |
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| Knowledge |
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| Knowledge of mathematics at the level of KMA/M1, KMT/LA (basics of differential calculus and linear algebra) |
| Skills |
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| solve more complex combinatorial problems |
| use classical probability skills |
| Competences |
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| N/A |
| N/A |
| N/A |
| N/A |
| learning outcomes |
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| Knowledge |
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| describe the properties of probability as a measure of random phenomena |
| understand descriptive parameters for random variables |
| define the concept of multidimensional random variables |
| define a random variable and understand its use in applications |
| understand the dependence of two or more random variables |
| describe known procedures for limit theorems |
| Skills |
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| analyzovat dané úlohy a řešit je s pomocí znalostí jednotlivých náhodných veličin |
| aplikovat teoretické znalosti z teorie pravděpodobnosti na konkrétní aplikační úlohy |
| nalézt vztahy mezi jednotlivými náhodnými veličinami |
| aplikovat varianty centrální limitní věty i věty o extremálních rozděleních |
| apply variants of the central limit theorem |
| Competences |
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| N/A |
| teaching methods |
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| Knowledge |
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| Practicum |
| Individual study |
| Interactive lecture |
| Přednáška s diskusí |
| Skills |
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| Seminární výuka |
| Practicum |
| Individual study |
| Interactive lecture |
| Competences |
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| self-study of students using texts on the portal and selected chapters from foreign literature |
| Skills demonstration |
| assessment methods |
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| Knowledge |
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| Combined exam |
| Test |
| Seminar work |
| Written and oral exam Demonstration of solved tasks at the blackboard, analysis of possible errors |
| Skills |
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| Combined exam |
| Test |
| Seminar work |
| Written and oral exam The student solves theoretical and practical tasks |
| Competences |
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| The student solves a set of problems, which he demonstrates in front of other classmates. He proceeds to the oral exam if he actively participates in the exercises and has successfully solved the given examples |
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Recommended literature
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Anděl, Jiří. Matematika náhody. Vyd. 2. Praha : Matfyzpress, 2003. ISBN 80-86732-07-X.
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C. M. Grinstead, J. L. Snell. Introduction to Probability. Boston, 1997. ISBN 0821807498.
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Dupač, Václav; Hušková, Marie. Pravděpodobnost a matematická statistika. Praha : Karolinum, 2001. ISBN 80-246-0009-9.
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E. T. Jaynes. Probability Theory: The Logic of Science. Cambridge, 2002. ISBN 0521592712.
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Kahounová, Jana. Praktikum k výuce matematické statistiky I : odhady. Praha : Vysoká škola ekonomická, 2000. ISBN 80-245-0070-1.
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Meloun, Milan; Militký, Jiří. Statistické zpracování experimentálních dat : v chemometrii, biometrii, ekonometrii a v dalších oborech přírodních, technických a společenských věd. 2. vyd. Praha : East Publishing, 1998. ISBN 80-7219-003-2.
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Plocki, A.; Tlustý, P. Pravděpodobnost a statistika pro začátečníky a mírně pokročilé. Praha, Prometheus, 2007. ISBN 9788071963301.
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Prasanna Sahoo. Probability and Mathematical Statistics. University of Louisville, 2013.
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Prášková Z., Lachout P. Základy náhodných procesů. Karolinum Praha, 1998.
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Riečan, Beloslav. Pravdepodobnosť a matematická štatistika. 1. vyd. Bratislava : Alfa, 1984.
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Štěpán, Josef. Teorie pravděpodobnosti : Matematické základy : Vysokošk. učebnice pro stud. matematicko-fyz. fakult. Praha : Academia, 1987.
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ZVÁRA, K., ŠTĚPÁN, J. Pravděpodobnost a matematická statistika. Vyd. 3. Praha : Matfyzpress, 2002. ISBN 80-85863-93-6.
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