Outcomes, events, probabilities, conditional probability, independence of events.Random variables, the distribution of random variables, basic examples of discrete distributions, continuous distributions, multi-dimensional distributions, conditional distribution in the discrete case, the independence of random variables, the distribution of sums of random variables.

The expectation of random variables, the variance of random variables, the expectation of functions of random variables, the covariance, the conditional expectation.Convergence of random variables, the weak law of large numbers, convergence in distribution, the central limit theorem, applications of the central limit theorem.

# Probability

G. Grimmett, D. Welsh: Probability: An Introduction, Oxford Science Publications, 1986.

J. Pitman, Probability. Springer Verlag, 1992.

D. Stirzaker, Probability: a beginner’s guide, Cambridge University Press, 1999.

Students will acquire the basic concepts of probability theory, which are starting points for many uses of mathematics.

Knowledge and understanding:

The course introduces the basic concepts of probability theory, which should be part of mathematics education. The emphasis is on the indispensable core concepts necessary to understand the statistics and financial mathematics.

Application:

The concepts of probability are used in most areas of science and social science.

The probability is a starting point for most financial mathematics.

Reflection: The course is leaning on other fields such as: analysis, discrete structures and linear algebra, while increasing the sense of the use of mathematics.

Transferable skills: Concepts and thinking in probabilities are the starting point for mathematical modeling.

Lectures, exercises, homeworks, consultations

2 midterm exams instead of written exam, written exam

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Roman Drnovšek:

DRNOVŠEK, Roman. Spectral inequalities for compact integral operators on Banach function spaces. Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, 1992, let. 112, str. 589-598. [COBISS-SI-ID 8169561]

DRNOVŠEK, Roman. On invariant subspaces of Volterra-type operators. Integral equations and operator theory, ISSN 0378-620X, 1997, let. 27, št. 1, str. 1-9. [COBISS-SI-ID 7038553]

DRNOVŠEK, Roman. A generalization of Levinger's theorem to positive kernel operators. Glasgow mathematical journal, ISSN 0017-0895, 2003, vol. 45, part 3, str. 545-555. [COBISS-SI-ID 12825945]

Mihael Perman:

PERMAN, Mihael. An excursion approach to Ray-Knight theorems for perturbed Brownian motion. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 1996, let. 63, str. 67-74. [COBISS-SI-ID 7621465]

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]

KOMELJ, Janez, PERMAN, Mihael. Joint characteristic functions construction via copulas. Insurance. Mathematics & economics, ISSN 0167-6687, 2010, vol. 47, iss. 2, str. 137-143. [COBISS-SI-ID 16242777]