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Lecturer(s)
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Course content
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FOR SUMMER SEMESTER OF SCHOOL YEAR 2024/2025 Probability measure. Outcomes of the random experiment, algebra and sigma-algebra of events. Finitely additive and sigma-additive probability, probability measure, probability space, examples. Random variables. Random variable with general state space and its distribution. Discrete and continuous distribution, density. Stochastic process. Stochastic process, product sigma-algebra, existence of the distribution of the process. Random vectors and sequences, process with values in R and continuous trajectories. Real random variable. Real random variable and vector. Distribution function, discrete, continuous and singular component. Mean and other moments. Characteristic function, relationship with moments. Convergence. Convergences of random variables: point, almost certainly, in probability, in the mean. Weak convergence of probability measures, convergence in distribution, convergence of distribution and chrakteristic functions. Mutual relationships, convergence of transformed variables. Independence. Independence of systems of events and random variables, product measure. Zero-one laws. Borel and Cantelli lemma. Tail and symmetric events, Kolmogorov and Hewitt-Savage zero-one law. The law of large numbers. Chebyshev's weak law of large numbers, strong law of large numbers for independent and identically distributed variables. Central limit theorem. Lindeberg-Levy central limit theorem, Feller-Lindeberg and Lyapunov condition. Conditional expected value. Definition of the conditional expected value, conditioning with respect to sigma-algebras and random variables, conditional density, conditional probability. Properties of the conditional expected value as an integral, taking out, independence, conditional expected valule as a projection. System of conditional distributions. Additional information on the web page http://home.zcu.cz/~friesl/Vyuka/Tp.html
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Learning activities and teaching methods
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Lecture with practical applications, Students' self-study, Self-study of literature, Textual studies, Lecture, Practicum
- Contact hours
- 52 hours per semester
- Preparation for formative assessments (2-20)
- 39 hours per semester
- Preparation for comprehensive test (10-40)
- 20 hours per semester
- Preparation for an examination (30-60)
- 50 hours per semester
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| prerequisite |
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| Knowledge |
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| formulovat a vysvětlit základní pojmy teorie míry a Lebesgueova integrálu (v rozsahu předmětu KMA/MA5) |
| formulovat a vysvětlit základní pojmy pravděpodobnosti a statistiky (v rozsahu předmětu KMA/PSA) |
| Skills |
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| pracovat s abstraktními strukturami teorie míry |
| vypočítat určité i neurčité integrály (známých typů) |
| využívat znalostí základních statistických metod a postupů pro jednoduchou analýzu dat |
| odlišit různé typy náhodných veličin (diskrétní, spojité) a různé typy rozdělení |
| Competences |
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| N/A |
| N/A |
| N/A |
| learning outcomes |
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| Knowledge |
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| orientovat se v probraných pojmech a výsledcích teorie pravděpodobnosti |
| Skills |
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| formulovat přesně matematicky probrané pojmy a výsledky teorie pravděpodobnosti |
| odvodit vyložené vlastnosti a vztahy |
| Competences |
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| N/A |
| N/A |
| teaching methods |
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| Knowledge |
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| Lecture |
| Interactive lecture |
| Self-study of literature |
| Practicum |
| Textual studies |
| Skills |
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| Interactive lecture |
| Self-study of literature |
| Lecture |
| Textual studies |
| Practicum |
| Competences |
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| Textual studies |
| Lecture |
| Practicum |
| Interactive lecture |
| Self-study of literature |
| assessment methods |
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| Knowledge |
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| Oral exam |
| Skills demonstration during practicum |
| Written exam |
| Skills |
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| Skills demonstration during practicum |
| Written exam |
| Oral exam |
| Competences |
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| Skills demonstration during practicum |
| Written exam |
| Oral exam |
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Recommended literature
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Kallenberg, Olav. Foundations of modern probability. 2nd ed. New York : Springer, 2002. ISBN 0-387-95313-2.
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Lachout, Petr. Teorie pravděpodobnosti. Praha : Karolinum, 2004. ISBN 80-246-0872-3.
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Štěpán, Josef. Teorie pravděpodobnosti : Matematické základy : Vysokošk. učebnice pro stud. matematicko-fyz. fakult. Praha : Academia, 1987.
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