Course: Fundamentals of Random Processes

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Course title Fundamentals of Random Processes
Course code KMA/ZNP
Organizational form of instruction Lecture + Tutorial
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 6
Language of instruction Czech, English
Status of course Compulsory, Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Course availability The course is available to visiting students
  • Sobotka Tomáš, Mgr. M.Sc.
  • Pospíšil Jan, Ing. Ph.D.
  • Tomiczková Světlana, RNDr. Ph.D.
Course content
1. Random number generators and their properties. Analysis of random data. Principles of Monte Carlo methods. 2. Definitions and basic characteristics of random processes. Classification. Examples. 3. Definition and elementary properties of discrete-time Markov chains (DTMC). Classification of states. Stationary and limit distributions. 4. DTMC examples: random walk, gambler's ruin, branching processes. discrete population models, processes with weighted transitions. 5. Definition and elementary properties of Markov chains with general state space. Examples. 6. Markov chains Monte Carlo (MCMC) methods, perfect simulations, their properties and applications. 7. Definition and elementary properties of continuous-time Markov chains (CTMC). Classification of states. 8. Kolmogorov differential equations and their solution. Stationary and limit distributions. 9. CTMC examples: Poisson process, growth processes, birth and death processes, queueing systems, renewal processes.

Learning activities and teaching methods
Interactive lecture, Lecture supplemented with a discussion, Lecture with practical applications, Students' portfolio, Task-based study method, Individual study, Students' self-study
  • Preparation for an examination (30-60) - 50 hours per semester
  • Individual project (40) - 40 hours per semester
  • Contact hours - 65 hours per semester
professional knowledge
Students should have a basic knowledge of probability theory (KMA/PSA).
learning outcomes
Students taking this course will be able to grasp the fundamentals of random processes and namely: - recognize and classify Markov chain with discrete and continuous time and name their basic properties, - generate random numbers with given properties, - apply Markov chains and Monte Carlo methods to practical problems, especially in statistical physics, economics and finance, - analyze Poisson process, growth, birth and death processes, queing systems and renewal processes, - provide logical and coherent proofs of theoretic results, - solve problems via abstract methods, - apply correctly formal and rigorous competency in mathematical presentation, both in written and verbal form.
teaching methods
Lecture supplemented with a discussion
Interactive lecture
Task-based study method
Students' self-study
Individual study
Students' portfolio
Lecture with practical applications
assessment methods
Oral exam
Written exam
Seminar work
Individual presentation at a seminar
Recommended literature
  • Brémaud, Pierre. Markov chains : Gibbs fields, Monte Carlo simulation, and queues. New York : Springer, 1999. ISBN 0-387-98509-3.
  • Häggström, Olle. Finite Markov chains and algorithmic applications. Cambridge . Cambridge University Press, 2002. ISBN 0-521-89001-2.
  • Havrda, Jan. Náhodné procesy. dot. 1. vyd. Praha : ČVUT, 1980.
  • Mandl, Petr. Pravděpodobnostní dynamické modely : celost. vysokošk. učebnice pro stud. matematicko-fyz. fakult stud. oboru pravděpodobnost a matem. statistika. Praha : Academia, 1985.
  • Prášková, Zuzana; Lachout, Petr. Základy náhodných procesů. Praha : Karolinum, 1998. ISBN 80-7184-688-0.
  • Stewart, William J. Introduction to the numerical solution of Markov chains. Princeton : Princeton University Press, 1994. ISBN 0-691-03699-3.

Study plans that include the course