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Course info
KMA / TP
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Course description
Department/Unit / Abbreviation
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KMA
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TP
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Probability Theory
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
5
Cred.
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Type of completion
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Combined
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Type of completion
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Combined
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Time requirements
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Lecture
3
[Hours/Week]
Tutorial
1
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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Czech
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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3 / -
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0 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Summer semester
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Semester taught
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Summer semester
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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N/A
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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To establish the theory of probability, to define related terms, to formulate and derive some of their properties and results - Probability measure, random variable (vector, sequence, process), distribution. Real random variable, expected value, characteristic function. Convergence of random variables and probability measures. Independence, 0-1 laws, law of large numbers, central limit theorem. Conditional expectation.
The course extends the material of the course KMA/PSA (introductory course of probability and statistics), and requires knowledge of KMA/MA5 (Measure and Integral Theory).
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Requirements on student
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Knowledge of the material and ability to apply it.
Upon repeated registration of the course, the credit obtained in the previous study of this course is not recognized.
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Content
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FOR SUMMER SEMESTER OF SCHOOL YEAR 2023/2024
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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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Contact hours
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52
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Preparation for an examination (30-60)
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50
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Preparation for comprehensive test (10-40)
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20
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Preparation for formative assessments (2-20)
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39
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Total
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161
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
formulovat a vysvětlit základní pojmy pravděpodobnosti a statistiky (v rozsahu předmětu KMA/PSA) |
formulovat a vysvětlit základní pojmy teorie míry a Lebesgueova integrálu (v rozsahu předmětu KMA/MA5) |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
odlišit různé typy náhodných veličin (diskrétní, spojité) a různé typy rozdělení |
pracovat s abstraktními strukturami teorie míry |
vypočítat určité i neurčité integrály (známých typů) |
využívat znalostí základních statistických metod a postupů pro jednoduchou analýzu dat |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
orientovat se v probraných pojmech a výsledcích teorie pravděpodobnosti |
Skills - skills resulting from the course: |
formulovat přesně matematicky probrané pojmy a výsledky teorie pravděpodobnosti |
odvodit vyložené vlastnosti a vztahy |
Competences - competences resulting from the course: |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Oral exam |
Skills demonstration during practicum |
Written exam |
Skills - skills achieved by taking this course are verified by the following means: |
Oral exam |
Skills demonstration during practicum |
Written exam |
Competences - competence achieved by taking this course are verified by the following means: |
Oral exam |
Skills demonstration during practicum |
Written exam |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Interactive lecture |
Lecture |
Practicum |
Self-study of literature |
Textual studies |
Skills - the following training methods are used to achieve the required skills: |
Interactive lecture |
Lecture |
Practicum |
Self-study of literature |
Textual studies |
Competences - the following training methods are used to achieve the required competences: |
Interactive lecture |
Lecture |
Practicum |
Self-study of literature |
Textual studies |
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