Probability and statistics

2022/2023
Programme:
Financial mathematics, First Cycle
Year:
2 year
Semester:
second
Kind:
mandatory
ECTS:
6
Language:
slovenian
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Prerequisites

Completed courses Analysis 2 and Probability 1.

Content (Syllabus outline)

Continuous random variables.
Joint continuous distributions.
Functions of continuous random verctors, transformation formula.
Convergence of distributions, the central limit theorem, law of large numbers.
Descriptive statistics and graphical presentation of the data.
Statistical models, parameters.
Sampling, sampling designs, estimators,sampling distribution, standard errors, normal approximation, confidence intervals, stratified sampling, ratio estimators.
Parameter estimation, estimators, unbiased estimators, maximum likelihood, asymptotic properties of estimators.
Hypothesis testing, statement of the problem, examples, test statistics, distribution of test statistics, size and power of the test, likelihood
ratio test.
Regression, regression model, least squares method, Gauss-Markov theorem, standard errors and hypothesis tests in regression, diagnostic methods.

Readings

D. Stirzaker, Probability and Random Variables, A beginner's guide, Cambridge University Press, 1999.
G. Grimmett and D. Stirzaker, Probability and Random Processes, Third Edition, Oxford University Press, 1982.
J Rice, Mathematical Statistics & Data Analysis, Third Edition, Duxburry, 2007.

Objectives and competences

Analysing and interpreting data is an essential part of the work of a financial mathematician. The course presents statistical concepts and statistical models most commonly used in statistical practice

Intended learning outcomes

Introduction of statistical concepts sufficient for independent study and the ability to present and analyze data with more advanced statistical models.

Learning and teaching methods

Lectures, problem sessions, seminar assignment.

Assessment

2 midterms or written exam, oral exam.
Seminar assignment.
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

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, WELLNER, Jon A. On the distribution of Brownian areas. Annals of applied probability, ISSN 1050-5164, 1996, let. 6, št. 4, str. 1091-1111. [COBISS-SI-ID 7101017]
PERMAN, Mihael, WELLNER, Jon A. An excursion approach to maxima of the Brownian bridge. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 2014, vol. 124, iss. 9, str. 3106-3120. [COBISS-SI-ID 17154393]
SMREKAR, Jaka. Turning a self-map into a self-fibration. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2014, vol. 167, str. 76-79. [COBISS-SI-ID 16943705]
Jaka Smrekar:
SMREKAR, Jaka. Homotopy type of space of maps into a K(G,n). Homology, homotopy, and applications, ISSN 1532-0073, 2013, vol. 15, no. 1, str. 137-149. [COBISS-SI-ID 16643929]
Jaka Smrekar:
SMREKAR, Jaka. Homotopy type of mapping spaces and existence of geometric exponents. Forum mathematicum, ISSN 0933-7741, 2010, vol. 22, no. 3, str. 433-456. [COBISS-SI-ID 15638105]