Undergraduate Course: Essentials in Analysis and Probability (MATH10047)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 10 (Year 4 Undergraduate)
||Availability||Available to all students
|Summary||The central topic of this course is measure theory. Measure theory is the foundation for advanced topics in Analysis and Probability.
The course will cover many of the following topics:
Random events, sigma-algebras, monotone classes.
Measurable spaces, random variables - measurable functions.
Measures, probability measures, signed measures.
Borel sets in R^d, Lebesgue measure. Caratheodory extension theorem.
Sequences of events and random variables, Borel-Cantelli lemma.
Distributions of random variables. Independence of random variables.
Integral of measurable functions - mathematical expectation,.
Moments of random variables, L_p spaces.
Convergence concepts of measurable functions.
Limit theorems for integrals.
Weak and strong laws of large numbers.
Completeness of L_p spaces.
Conditional expectation and conditional distribution of random variables.
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2021/22, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 5%, Examination 95%
There will be 5 assignments. Each assignment will be marked out of 20; a mark of 7 or lower on an assignment will be recorded as no credit, and a mark of 8 or higher will be recorded as full credit.
Some assignment questions are harder than others.
The assignment will be due by W3, W5, W7 W9 and W11. At the end of the semester, the best 4 out of 5 assignments will be counted.
||Feedback will be given by marking and commenting the assignments, by individual discussions during tutorials and office hours.
||Hours & Minutes
|Main Exam Diet S1 (December)||2:00|
On completion of this course, the student will be able to:
- use the basic notions and results from measure theory and integration.
- relate the language of measure to the fundamental concepts of probability theory.
- use the main results of this course to compute integrals and probabilities and to justify the applicability of those results and
- construct proofs for previously unseen results.
|R. M. Dudley, Real Analysis and Probability, Cambridge University Press, 2004. |
J. Jacod and P. Protter, Probability Essentials, Springer 2004
H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, third edition, 1988.
|Course organiser||Prof Istvan Gyongy
Tel: (0131 6)50 5945
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045