Syllabus:

# Advanced Probability Theory, 7.5 Credits

This syllabus is valid: 2021-01-18 and until further notice

Course code: 5MS076

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematical Statistics: Second cycle, has second-cycle course/s as entry requirements

Grading scale: VG Pass with distinction, G Pass, U Fail

Established by: Faculty Board of Science and Technology, 2020-08-20

## Contents

The course will give the student a deeper understanding of the foundations of probability theory. It provides important concepts, results and proofs in measure-theoretic probability theory with emphasis in statistics. It covers probability spaces and random elements, integration and differentiation, distributions and their characteristics, conditional expectations, asymptotic theory, together with a large number of exercises which includes many additional results.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding
• independently give a count of the foundations of probability theory from a measure-theoretic perspective
• thoroughly explain important results and properties for moments
• thoroughly describe theory for conditional distributions and expectation from a measure-theoretic perspective
• thoroughly explain, define and relate different types of convergences of distributions and probability measures, as well as general central limit theorems
Skills and abilities
• derive advanced probability-theoretic results of importance for statistical inference
• independently prove important theorems in probability theory
• independently solve advanced problems in probability theory
Judgment and approach
• perform mathematically stringent probability-theoretic reasoning
• critically apply central results in probability theory on typical problems within the field.

## Required Knowledge

The course requires a minimum of 90 ECTS including 7.5 ECTS in Probability Theory at second cycle or equivalent. Proficiency in English and Swedish equivalent to the level required for basic eligibility for higher studies

## Form of instruction

The teaching mainly consists of lectures and lessons.

## Examination modes

The course is examined by combination of a written home exam and an oral exam. On the written exam one of the following judgements is assigned: Fail (U), Pass (G) or Pass with distinction (VG). On the oral exam one of the following judgements is assigned: Fail (U) or Pass (G). For the whole course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG). In order to receive a passing grade on the course, all parts must be completed with a passing judgement. The course grade constitutes a summary of the results on all examination and is assigned only when all mandatory examination has been completed.

Deviations from the form of examination specified in the syllabus form can be made for a student who has been granted pedagogical support due to disability. Individual adaptation of the examination form will be considered based on the student's needs. The examination form is adapted within the framework of the expected learning outcomes of the course syllabus. At the request of the student, the course coordinator, in consultation with the examiner, must promptly decide on the adapted examination form. The decision must then be communicated to the student.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Credit transfers
Students are entitled to an assessment of whether previous education or equivalent knowledge and skills acquired in professional experience can be accredited for equivalent studies at Umeå University. Applications for credit transfers must be sent to Student Services/Degree Evaluation Office. More information on credit transfers can be found on Umeå University's student website, www.student.umu.se, and in the Higher Education Ordinance (Chapter 6). Rejected applications for credit transfers can be appealed (Higher Education Ordinance, Chapter 12) to the Higher Education Appeals Board. This applies regardless of whether the rejection relates to all or parts of the credit transfer application.

## Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics.

## Literature

### Valid from: 2021 week 34

Gut Allan.
2nd ed. 2013. : New York, NY : Springer New York : 2013 : XXV, 600 p. 13 illus. :
ISBN: 9781461447085
Mandatory
Search the University Library catalogue

Ash Robert B.
Real analysis and probability
New York : Academic P. : cop. 1972 : 476 s. :
ISBN: 0120652013
Search the University Library catalogue

Billingsley Patrick
Probability and measure
cop. 2012 : xii, s. :
ISBN: 9781118122372
Search the University Library catalogue

Chung Kai Lai
A course in probability theory
3. ed. : San Diego, Calif. : Academic Press : cop. 2001 : 419 s. :
ISBN: 0121741516
Search the University Library catalogue