Advanced Probability Theory, 7.5 Credits

Contents

The course will give the student a deeper understanding of the foundations of probability theory, such as probability theory from a measure-theoretic perspective, convergences of distributions and probability measures, and conditional expectations. During the course, important theorems, such as Borell-Cantelli lemma, Radon-Nikodym theorem, Fubini theorem, and general central limit theorems, will be investigated.

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, define and relate different types of convergences of distributions and probability measures
• thoroughly explain important results and properties for expectation
• thoroughly describe theory for conditional distributions and expectation from a measure-theoretic perspective

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 7,5 ECTS in Probability Theory on advance level or equivalent. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish 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 written home exam and in the form of an oral test. On the written exam one of the following grades is assigned: Fail (U), Pass (G) or Pass with distinction (VG). On the oral test one of the following grades 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 grade. The course grade constitutes a summary of the results on all examination and is assigned only when all mandatory examination has been completed.

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.