Probability

2022/2023
Programme:
Mathematics Education
Year:
3 year
Semester:
first
Kind:
mandatory
ECTS:
5
Language:
slovenian
Hours per week – 1. semester:
Lectures
2.47
Seminar
0
Tutorial
2.53
Lab
0
Prerequisites

Completed courses Analysis 1, Analysis 2a and Analysis 2b.

Content (Syllabus outline)

Basic concepts: outcomes, events, probability of events, probability axioms, the elementary conditional probability, independent events.Random variables: introductory examples, distribution of a random variable, discrete distributions, expectation, conditional expectation.Random vectors: distributions, expectation of functions of random vectors, conditional distribution and conditional expectation, independence, sums of discrete random variables.Generating functions: definition, properties, applications, branching processes, random walks.Continuous random variables and vectors: density, multi-dimensional density, expectation of functions of continuous random vectors, conditional distribution and conditional expectation, normal distribution. Characteristic function.Convergence of random variables: weak law of large numbers, convergence in distribution, the Poisson approximation.

Readings

G. Grimmett, D. Welsh: Probability : An Introduction, Oxford Univ. Press, Oxford, 1986.
J. Pitman: Probability, Springer, New York, 1999.
D. Stirzaker: Probability and Random Variables : A Beginner’s Guide, Cambridge Univ. Press, Cambridge, 1999.

Objectives and competences

The course is devoted to the basics of probability theory. We introduce the notions of events and their probabilities, with a special emphasis on the notion of random variables and their distributions. We continue with the concepts of expectation, conditional expectation and conditional distributions. We conclude with convergence of random variables.

Intended learning outcomes

Knowledge and understanding: Probability is a standard part of mathematical education, and, on the other hand, the starting point for applications in a wide range of disciplines from biology, economics, financial and actuarial mathematics. Knowledge of basic concepts of probability is a necessary part of education of any mathematician.
Application: The use of concepts of probability extends to many areas of science and social science.
Reflection: Understanding of theoretical concepts in many applications.
Transferable skills: The ability to identify the probability concepts in other sciences (physics, economics, finance, actuarial science, medicine, biology).

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

Type (examination, oral, coursework, project):
written exam
oral exam
Grading: 6-10 pass, 1-5 fail (according to the rules of University of Ljubljana)

Lecturer's references

DRNOVŠEK, Roman. An infinite-dimensional generalization of Zenger's lemma. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2012, vol. 388, iss. 2, str. 1233-1238. [COBISS-SI-ID 16214617]
DRNOVŠEK, Roman, OMLADIČ, Matjaž. Maximal dominated operator semigroups. Semigroup forum, ISSN 0037-1912, 2002, vol. 64, no. 3, str. 376-390. [COBISS-SI-ID 11620185]
DRNOVŠEK, Roman, KOŠIR, Tomaž, KRAMAR, Edvard, LEŠNJAK, Gorazd. Zbirka rešenih nalog iz verjetnostnega računa, (Izbrana poglavja iz matematike in računalništva, 37). Ljubljana: Društvo matematikov, fizikov in astronomov Slovenije, 1998. 195 str. ISBN 961-212-086-2. [COBISS-SI-ID 75872256]
PERMAN, Mihael, WERNER, Wendelin. Perturbed Brownian motions. Probability theory and related fields, ISSN 0178-8051, 1997, let. 108, št. 3, str. 357-383. [COBISS-SI-ID 7848537]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities and decompositions for general perturbed risk processes. Annals of applied probability, ISSN 1050-5164, 2004, vol. 14, no. 3, str. 1378-1397. [COBISS-SI-ID 13168985]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities for competing claim processes. Journal of Applied Probability, ISSN 0021-9002, 2004, vol. 41, no. 3, str. 679-690. [COBISS-SI-ID 13207641]