Lecturer(s)


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 discretetime 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 continuoustime 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, Taskbased study method, Individual study, Students' selfstudy
 Preparation for an examination (3060)
 50 hours per semester
 Individual project (40)
 40 hours per semester
 Contact hours
 65 hours per semester

prerequisite 

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 
Taskbased study method 
Students' selfstudy 
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 0387985093.

Häggström, Olle. Finite Markov chains and algorithmic applications. Cambridge . Cambridge University Press, 2002. ISBN 0521890012.

Havrda, Jan. Náhodné procesy. dot. 1. vyd. Praha : ČVUT, 1980.

Mandl, Petr. Pravděpodobnostní dynamické modely : celost. vysokošk. učebnice pro stud. matematickofyz. fakult stud. oboru pravděpodobnost a matem. statistika. Praha : Academia, 1985.

Prášková, Zuzana; Lachout, Petr. Základy náhodných procesů. Praha : Karolinum, 1998. ISBN 8071846880.

Stewart, William J. Introduction to the numerical solution of Markov chains. Princeton : Princeton University Press, 1994. ISBN 0691036993.
