## 4H: Measure Theory and Probability MATHS4116

• School: School of Mathematics and Statistics
• Credits: 10
• Level: Level 4 (SCQF level 10)
• Typically Offered: Semester 1
• Available to Visiting Students: Yes

### Short Description

This course introduces measure theory in a rigorous way and explores some applications to probability theory. Both of these are core mathematical disciplines. In addition, the knowledge of probability theory provided in this course is essential basis for further courses in mathematical finance.

### Timetable

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

None.

### Excluded Courses

Probability STATS4060

None

### Assessment

Assessment

90% Examination, 10% Coursework.

Main Assessment In: April/May

Are reassessment opportunities available for all summative assessments? Not applicable for Honours courses

Reassessments are normally available for all courses, except those which contribute to the Honours classification. For non-Honours courses, students are offered reassessment in all or any of the components of assessment if the satisfactory (threshold) grade for the overall course is not achieved at the first attempt. This is normally grade D3 for undergraduate students and grade C3 for postgraduate students. Exceptionally it may not be possible to offer reassessment of some coursework items, in which case the mark achieved at the first attempt will be counted towards the final course grade. Any such exceptions for this course are described below.

### Course Aims

The aim of this course is to introduce students to measure theory in a rigorous way, and explore some applications to probability theory.

### Intended Learning Outcomes of Course

By the end of this course students will be able to:

Demonstrate knowledge of the central definitions and facts of measure theory and probability theory and use these to solve problems of a numerical or logical nature. In particular, students will be able to

■ Explain the notion of and give definitions of measure space and probability space and list their basic properties.

■ Integrate simple positive measurable functions, positive measurable functions, and general L1 -functions and derive properties of integrals this way.

■ State and apply the monotone convergence theorem, dominated convergence theorem, and the Fubini theorem.

■ Determine whether two or more random variables are independent.

### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.