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Lecturer(s)
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Tomovski Zhivorad, prof. Ph.D.
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Course content
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1. Characteristic functions, moment generating function and their connections 2. Central limit theorem, proof using characteristic functions 3. Inversion formula for characteristic functions 4. Moments of random variables, kurtosis and skewness 5. Weak convergence of random variables, method of moments 6. Series of random variables, convergence and problem of 3 series 7. Theorem of Bochner-Khinchin and Polya proof and examples 8. N-dimensional normal distribution, special case N=2 and examples 9. Conditional mathematical expectation and some properties 10. Spaces L2 (Ω), Lp (Ω), p>1 of random variables, scalar product of random variables and geometrical properties 11. Information, Entropy of random variables and examples. 12. Random processes, Brownian motion and Wiener process 13. Markov process and Poisson process
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Learning activities and teaching methods
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unspecified
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Learning outcomes
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Gain knowledge about the theory of characteristic functions, examples, exercises, moments and their applications, to define Hilbert and Banach space of random variables, to present some geometric properties and structures, introduction and analysis of some stochastic processes.
Gain useful knowledge about theory of random vectors, weak convergence, characteristic functions of random variables, Information and Entropy, introduction of stochastic processes and examples like Brownian motion, Markov process etc.
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Prerequisites
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Student has to pass the basic courses of Probability and Statistics. Standard knowledge from mathematical analysis and linear algebra. Elemental experience with computation on PC
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Assessment methods and criteria
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unspecified
Credit: The student has to compute given examples Exam: Combination of oral and written exam. The student has to understand the subject and be acquainted with the theoretical and practical aspects of the methods
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Recommended literature
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Durret, R. (2010). Probability: Theory and Examples.
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Knill, O. (2009). Probability and Stochastic Processes with Applications.
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Korosteleva, O. (2022). Stochastic processes with R, An introduction.
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Parzen, E. (1999). Stochastic Processes, Society for Industrial and Applied Mathematics.
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Scott, M. (2012). Probability and Random processes.
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