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An introductory course in probability theory and statistics. Required course for General Mathematics.
Last update: Zichová Jitka, RNDr., Dr. (26.04.2018)
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Foundations of probability theory, mathematical statistics and principles of stochastic thinking Last update: Pešta Michal, doc. RNDr., Ph.D. (15.02.2024)
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To complete the course, it is necessary to obtain credit for the exercises and successfully pass the exam.
Credit for the exercises is a prerequisite for participation in the exam and registration for it.
Conditions for obtaining credit:
A credit will be obtained by those who:
Last update: Pešta Michal, doc. RNDr., Ph.D. (26.05.2025)
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[1] Casella, G. and Berger, R.L. (2001) Statistical Inference, 2nd Edition. Pacific Grove, CA: Duxbury
[2] Chung, K.L. (2001) A Course in Probability Theory, 3rd Edition. San Diego, CA: Academic Press
[3] Dupač, V. and Hušková, M. (2013) Pravděpodobnost a matematická statistika. Praha, CZ: Karolinum
[4] Resnick, S.I. (2013) A Probability Path, 2014th Edition. Basel, CH: Birkhäuser
[5] Rosenthal, J.S. (2006) A First Look at Rigorous Probability Theory, 2nd Edition. Singapore, SG: World Scientific
[6] Ross, S.M. (2020) A First Course in Probability, 10th Edition. London, UK: Pearson
[7] Wasserman, L. (2013) All of Statistics: A Concise Course in Statistical Inference. New York, NY: Springer Last update: Pešta Michal, doc. RNDr., Ph.D. (15.02.2024)
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Lecture & exercises. Last update: Pešta Michal, doc. RNDr., Ph.D. (15.02.2024)
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A prerequisite for registering for and participating in the exam is obtaining a credit.
The subject of the exam will be the entire (presented) scope of the lecture. It is necessary to know all essential definitions, theorems and statements (including assumptions), understand their mutual relationships and at least broadly explain their justification (proofs).
The exam consists of a written and oral part. The written part precedes the oral part and failure to pass it means that the entire exam is assessed with a failed grade and the oral part is no longer continued. Failure to pass the oral part of the exam means that both parts of the exam, i.e., the written and oral, must be repeated at the next date. The exam grade is determined on the basis of the written and oral parts.
The requirements for the oral part of the exam correspond to the subject syllabus in the scope that was presented at the lecture. Last update: Pešta Michal, doc. RNDr., Ph.D. (26.05.2025)
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Basic concepts of probability theory: classical and axiomatic definition of probability, conditional probability, independence of random events, the law of total probability, Bayes' theorem.
Random variables: Definition of a random variable, its distribution and its distribution function, their properties, discrete and continuous distributions, mean value and variance of a random variable, other numerical characteristics of random variables, distribution of functions of random variables.
Random vectors: Definition of a random vector, its distribution and its distribution function, independence of random variables, numerical characteristics of random vectors, distribution of functions of random vectors.
Conditional distribution and conditional expectation. Transformation of random variables and random vectors. Characteristic function and moment generating function.
Stochastic inequalities: Chebyshev's inequality, Markov's inequality, Hoeffding's inequality, Mill's inequality, Cauchy-Schwartz inequality, Jensen's inequality.
Stochastic convergence: Convergence in probability, convergence in distribution, convergence in L2.
Limit theorems: Weak law of large numbers, central limit theorem, delta method.
Statistics: Foundations and basic concepts of statistics, random sample.
Parametric models: Point and interval estimation. Unbiased, consistent estimates. Method of moments, maximum likelihood method. Overview of basic interval estimation (based on normality and CLV).
Hypothesis testing: Formulation of statistical hypotheses, type I error, type II error, significance level, p-value.
Empirical distribution function. Statistical functionals. Bootstrap. Last update: Pešta Michal, doc. RNDr., Ph.D. (16.02.2024)
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Fundamentals of differential and integral calculus, fundamentals of linear algebra, fundamentals of measure theory. Last update: Pešta Michal, doc. RNDr., Ph.D. (26.05.2025)
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