Probability Theory, 7.5 Credits

Contents

The course covers random variables in one and several dimensions, conditional probabilities, probability generating and characteristic functions. The multivariate normal distribution and distributions of order statistics and quadratic forms are treated. Other important probabilistic results that are covered more thoroughly compared to courses on the Bachelor level, are covergence criteria for series of random variables, the Borel Cantelli lemma, convergence through transforms, the law of large numbers, the central limit theorem and Cramér-Slutsky's theorem. Finally a deepend presentation of Poisson processes is given.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• thoroughly account for important notions in probability theory
• prove some of the most important theorems, e.g. the central limit theorem and the law of large numbers
• derive results from probability theory that are important for statistical theory, such as distributions for order statistics and quadratic forms of normally distributed random variables
• thoroughly account for different concergence criteria, and describe how they relate to each other

Skills

• set up and analyze probabilistic models, by using advanced methods such as characteristic functions and conditional expectations
• apply different convergence criteria, and describe how they relate to each other

Required Knowledge

The course requires 90 ECTS including courses in Mathematical Statistics minimum 12 ECTS or courses in Statistics, minimum 75 ECTS and in both cases a course in Basic Calculus, 7,5 ECTS. Proficiency in English and Swedish equivalent to the level required for basic eligibility for higher studies.

Form of instruction

The teaching is in the form of lectures, group sessions and supervision.

Examination modes

The course is assessed through a written exam. During the course it is possible to get bonus points from written home assignments, that are added to the points on the written exam. The maximum bonus corresponds to 15% of the total sum of points on the written exam. The bonus points are valid on the first and second exams on the course occasion where the bonus points are awarded. For the course, one of the following grades is awarded: Fail (U), Pass (G), or Pass with distinction (VG), and the grade is decided from the result on the written exam.

Deviations from the syllabus examination form can be made for a student who has a decision on pedagogical support due to disability. Individual adaptation of the examination form shall 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 shall 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 transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

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. The course can also be included in the subject area of computational science and engineering.