Probability and Statistics

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Probability and Statistics (Main course) BE5B01PRS
Credits 7
Semesters Winter
Completion Assessment + Examination
Language of teaching English
Extent of teaching 4P+2S
Annotation
Introduction to the theory of probability, mathematical statistics and computing methods together with their applications of praxis.
Study targets
The aim is to introduce the students to the theory of probability and mathematical statistics, and show them the computing methods together with their applications of praxis.
Course outlines
1. Random events, probability, probability space.
2. Conditional probability, Bayes' theorem, independent events.
3. Random variable - definition, distribution function.
4. Characteristics of random variables.
5. Discrete random variable - examples and usage.
6. Continuous random variable - examples and usage.
7. Independence of random variables, sum of independent random variables.
8. Transformation of random variables.
9. Random vector, covariance and correlation.
10. Central limit theorem.
11. Random sampling and basic statistics.
12. Point estimation, method of maximum likelihood and method of moments, confidence intervals.
13. Confidence intervals and hypotheses testing.
14. Markov chains.
Exercises outlines
1. Random events, probability, probability space.
2. Conditional probability, Bayes' theorem, independent events.
3. Random variable - definition, distribution function.
4. Characteristics of random variables.
5. Discrete random variable - examples and usage.
6. Continuous random variable - examples and usage.
7. Independence of random variables, sum of independent random variables.
8. Transformation of random variables.
9. Random vector, covariance and correlation.
10. Central limit theorem.
11. Random sampling and basic statistics.
12. Point estimation, method of maximum likelihood and method of moments, confidence intervals.
13. Confidence intervals and hypotheses testing.
14. Markov chains.
Literature
[1] Papoulis, A.: Probability and Statistics, Prentice-Hall, 1990.
[2] Stewart W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press 2009.
Requirements
Basic calculus, namely integrals.
Mathematics for Economy BE1M01MEK
Credits 6
Semesters Winter
Completion Assessment + Examination
Language of teaching English
Extent of teaching 4P+2S
Annotation
The aim is to introduce basics of probability, statistics and random processes, especially with Markov chains, and show applications of these mathematical tools in economics.
Course outlines
1. Random events, probability, probability space, conditional probability, Bayes theorem, independent events.
2. Random variable - construction and usage of distribution function, probability function and density, characteristics of random variables - expected value, variance.
3. Discrete random variable - examples and usage.
4. Continuous random variable - examples and usage.
5. Independence of random variables, covariance, correlation, transformation of random variables, sum of independent random variables (convolution).
6. Random vector, joint and marginal distribution, central limit theorem.
7. Random sampling and basic statistics, point estimates, maximum likelihood method and method of moments.
8. Confidence intervals.
9. Hypotheses testing.
10. Random processes - basic terms.
11. Markov chains with discrete time - properties, transition probability matrix, classification of states.
12. Markov chains with continuous time - properties, transition probability matrix, classification of states.
13. Practical use of random processes - Wiener process, Poisson process, applications.
14. Linear regression.
Exercises outlines
1. Random events, probability, probability space, conditional probability, Bayes theorem, independent events.
2. Random variable - construction and usage of distribution function, probability function and density, characteristics of random variables - expected value, variance.
3. Discrete random variable - examples and usage.
4. Continuous random variable - examples and usage.
5. Independence of random variables, covariance, correlation, transformation of random variables, sum of independent random variables (convolution).
6. Random vector, joint and marginal distribution, central limit theorem.
7. Random sampling and basic statistics, point estimates, maximum likelihood method and method of moments.
8. Confidence intervals.
9. Hypotheses testing.
10. Random processes - basic terms.
11. Markov chains with discrete time - properties, transition probability matrix, classification of states.
12. Markov chains with continuous time - properties, transition probability matrix, classification of states.
13. Practical use of random processes - Wiener process, Poisson process, applications.
14. Linear regression.
Literature
[1] Papoulis, A.: Probability and Statistics, Prentice-Hall, 1990.
[2] Stewart W.J.: Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press 2009.
[3] Kaas, R., Goovaerts, M., Dhaene, J., Denuit, M.: Modern actuarial risk theory. Kluwer Academic Publishers, 2004.
[4] Gerber, H.U.: Life Insurance Mathematics. Springer-Verlag, New York-Berlin-Heidelberg, 1990.
[5] Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification. John Wiley & Sons, 2001.