This module runs in Term 1 and aims to provide an introduction to this theory, concentrating on examples and applications. This course would particularly be useful for students willing to learn more about probability theory, analysis, mathematical finance, and theoretical statistics.
This module is available for students on a course where it is an optional core module or listed option and as an Unusual Option to students who have completed the prerequisite modules.
Pre-requisites:
Statistics Students: ST218 Mathematical Statistics A AND ST219 Mathematical Statistics B
Non-Statistics Students: ST220 Introduction to Mathematical Statistics
Leads to: ST318 Probability Theory.
To introduce the concepts of measurable spaces, integral with respect to the Lebesgue measure, independence and modes of convergence, and provide a basis for further studies in Probability, Statistics and Applied Mathematics. Imagine picking a real number x between 0 and 1 "at random" and with perfect accuracy, so that the probability that this number belongs to any interval within [0,1] is equal to the length of the interval. Can we compute the probability of x belonging to any subset to [0,1]?
To answer this question rigorously we need to develop a mathematical framework in which we can model the notion of picking a real number "at random". The mathematics we need, called measure theory, permeates through much of modern mathematics, probability and statistics.
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
I. Algebras, sigma-algebras and measures
Algebra and contents, sigma-algebra and measures, pi-systems, examples of random events and measurable sets.
II. Lebesgue integration
Simple functions, standard representations, measurable functions, Lebesgue integral, properties of integrals, integration of Borel functions.
III. Product measures, 2 lectures
Sections, product sigma-algebras, product measures, Fubini theorem.
IV. Independence and conditional expectation 3 lectures
Independence of sigma-algebras, independence of random variables, conditional expectation with respect to a simple algebra.
V. Convergence and modes of convergence
Borel-Cantelli lemma, Fatou lemma, dominated convergence theorem, modes of convergence of random variables, Markov inequality and application, weak and strong laws of large numbers.
By the end of the module, students should be able to:
View reading list on Talis Aspire
TBC
TBC
| Type | Required |
|---|---|
| Lectures | 30 sessions of 1 hour (20%) |
| Tutorials | 5 sessions of 1 hour (3%) |
| Private study | 115 hours (77%) |
| Total | 150 hours |
Weekly revision of lecture notes and materials, wider reading, practice exercises and preparing for examination.
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| In-person Examination | 100% | No | |
|
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
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| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| In-person Examination - Resit | 100% | No | |
|
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
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Solutions and cohort level feedback will be provided for the examination. The results of the January examination will be available by the end of week 10 of term 2.
If you take this module, you cannot also take:
This module is Core optional for:
This module is Optional for:
This module is Core option list B for:
This module is Option list A for:
This module is Option list C for: