Undergraduate Course: Essentials in Analysis and Probability (MATH10047)
Course Outline
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 
SCQF Credits  10 
ECTS Credits  5 
Summary  The central topic of this course is measure theory. Measure theory is the foundation for advanced topics in Analysis and Probability.

Course description 
The course will cover many of the following topics:
Random events, sigmaalgebras, 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, BorelCantelli 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.
Fubini's theorem.

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2020/21, Available to all students (SV1)

Quota: None 
Course Start 
Semester 1 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %

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 
Feedback will be given by marking and commenting the assignments, by individual discussions during tutorials and office hours. 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)   2:00  
Learning Outcomes
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.

Reading List
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. 
Contacts
Course organiser  Prof Istvan Gyongy
Tel: (0131 6)50 5945
Email: I.Gyongy@ed.ac.uk 
Course secretary  Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk 

