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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Essentials in Analysis and Probability (MATH10047)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 10 (Year 4 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryThe 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, 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.
Fubini's theorem.
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Honours Complex Variables (MATH10067) AND Honours Differential Equations (MATH10066) AND Honours Analysis (MATH10068)
Prohibited Combinations Other requirements None
Information for Visiting Students
High Demand Course? Yes
Course Delivery Information
Academic year 2023/24, 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:
  1. use the basic notions and results from measure theory and integration.
  2. relate the language of measure to the fundamental concepts of probability theory.
  3. use the main results of this course to compute integrals and probabilities and to justify the applicability of those results and
  4. 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.
Additional Information
Graduate Attributes and Skills Not entered
Course organiserProf Istvan Gyongy
Tel: (0131 6)50 5945
Course secretaryMrs Alison Fairgrieve
Tel: (0131 6)50 5045
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