Introductory description

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 module provides a strong foundation to the measure theory underpinning probability, concentrating on examples and applications. This module 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: ST228 Mathematical Methods for Statistics and Probability AND ST229 Probability for Mathematical Statistics AND ST230 Mathematical Statistics

• Non-Statistics Students: ST232/233 Introduction to Mathematical Statistics

Leads to: ST318 Probability Theory and other advanced probability modules.

Module web page

Module aims

Outline syllabus

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.

1. Algebras, sigma-algebras and measures
   Algebra and contents, sigma-algebra and measures, pi-systems, examples of random events and measurable sets.

2. Lebesgue integration
   Simple functions, standard representations, measurable functions, Lebesgue integral, properties of integrals, integration of Borel functions.

3. Product measures.
   Sections, product sigma-algebras, product measures, Fubini theorem.

4. Independence and conditional expectation.
   Independence of sigma-algebras, independence of random variables, conditional expectation with respect to a simple algebra.

5. Convergence and modes of convergence
   Borel-Cantelli lemma, Fatou's lemma, dominated convergence theorem, modes of convergence of random variables, Markov's inequality and application, weak and strong laws of large numbers.

Learning outcomes

By the end of the module, students should be able to:

• Apply properties of the probability spaces to build models for random experiments.
• Evaluate measure-theoretic techniques and apply them to determine probabilities of events.
• Rigorous formulate and apply formal notions of probability, including computing probability, statistical independence and expectation, to a range of situations.
• Apply measure-theoretic integration to a range of situations to derive results regarding random variables from first principles.
• Apply modes of convergence of sequences of random variables to a breath of situations in probability and statistics.
• Apply and justify convergence in the computation of integrals and expectations.

Indicative reading list

Donald Cohn, (2010), Measure Theory, 2nd Edition, Birkhauser

View reading list on Talis Aspire

Subject specific skills

• Demonstrate facility with rigorous probabilistic methods.

• Evaluate, select and apply appropriate mathematical and/or probabilist techniques.

• Demonstrate knowledge of and facility with formal probability concepts, both explicitly and by applying them to the solution of mathematical problems.

• Create structured and coherent arguments communicating them in written form. 

• Construct logical mathematical arguments with clear identification of assumptions and conclusions.

• Reason critically, carefully, and logically and derive (prove) mathematical results.

Transferable skills

• Problem solving: Use rational and logical reasoning to deduce appropriate and well-reasoned conclusions. Retain an open mind, optimistic of finding solutions, thinking laterally and creatively to look beyond the obvious. Know how to learn from failure.

• Self awareness: Reflect on learning, seeking feedback on and evaluating personal practices, strengths and opportunities for personal growth.

• Communication: Written: Present arguments, knowledge and ideas, in a range of formats.

• Professionalism: Prepared to operate autonomously. Aware of how to be efficient and resilient. Manage priorities and time. Self-motivated, setting and achieving goals, prioritising tasks.