Stochastic Modelling

Data is displayed for academic year: 2023./2024.

Lecturers

Assoc. Prof.
Lenka Mihoković
Asst. Prof.
Stjepan Šebek

Course Description

In the first part of the course, the generation of pseudorandom numbers will be studied. Most of the probability distributions that appear in applications will be covered, and many interesting relations between them will be observed, and used to produce the pseudorandom numbers from those distributions. The codes for simulating values from different distributions will be written using a computer (R programming language will be used). The second part is concerned with Monte Carlo simulations, with a focus on Monte Carlo integration, and methods for reducing the variance of the Monte Carlo estimates. The codes for evaluating integrals using simulations will again be written in R. In the last part, concept of a random process will be introduced. Random walk will be studied as an example of discrete time random process, and Brownian motion will be studied as an example of continuous time random process. Both random walk and Brownian motion will be simulated, and those simulations will be used to approximately compute some values whose exact value is still unknown in the literature.

Study Programmes

University graduate
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Learning Outcomes

  1. Improve knowledge about the basic laws of random events
  2. Learn to distinguish the most important random variables according to their properties
  3. Model important distributions
  4. Model complex situations in which randomness occurs
  5. Model stochastic processes and Markov chains
  6. Apply the basic techniques of the Monte Carlo method

Forms of Teaching

Lectures

During lectures all the necessary results will be proved on the blackboard and immediately after that corresponding implementations will be done using computer (R programming language)

Seminars and workshops

Seminars are optional, and allowed topics are anything related to the course material.

Independent assignments

During the lectures, students will independently solve some tasks using computer and they will be rewarded in case of exceptional engagement. Furthermore, students will from time to time have some homework that they will have to solve using computer.

Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Homeworks 0 % 10 % 0 % 0 %
Seminar/Project 0 % 10 % 0 % 0 %
Mid Term Exam: Written 0 % 50 % 0 %
Final Exam: Written 0 % 50 %
Exam: Written 0 % 100 %
Comment:

Exams are written using a computer (typical problem is to write a code for some sort of simulation). Seminar and homework are optional for extra credit.

Week by Week Schedule

  1. Pseudorandom number generators. Statistical analysis of simulated data, goodness of fit tests.
  2. Inverse transformation method. Exponential distribution, logistic distribution, Cauchy distribution.
  3. General transformation methods, gamma distribution, chi-square distribution.
  4. Polar method (Box-Muller transform) for generating a normal random variable.
  5. Accept-reject method, beta distribution.
  6. Generation of discrete distributions. Geometric distribution.
  7. Binomial distribution, Poisson distribution.
  8. Midterm exam
  9. Introduction to Monte Carlo methods, use of random variables in the calculation of integrals.
  10. Importance sampling in Monte Carlo integration
  11. Simulating random walks.
  12. Using simulations of random walks in solving known problems from the theory of random processes.
  13. Simulating Brownian motion.
  14. Using simulations of Brownian motion in solving known problems from the theory of random processes.
  15. Final exam

Literature

(.), Neven Elezović, Stohastičko modeliranje - skripta,
(.), Nedžad Limić, Monte Carlo simulacije slučajnih veličina, nizova i procesa, Zagreb, Element 2002,,
(.), M. A. Pinsky, S. Karlin, An Introduction to Stochastic Modeling, Elsevier 2011,
(.), S. M. Ross, Introduction to Probability Models, 10th ed., Elsevier 2010,

For students

General

ID 222785
  Winter semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures
0 Seminar
0 Exercises
0 Laboratory exercises
0 Project laboratory
0 Physical education excercises

Grading System

85 Excellent
70 Very Good
55 Good
45 Sufficient

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