Introductory description

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.

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 lemma, dominated convergence theorem, modes of convergence of random variables, Markov inequality and application, weak and strong laws of large numbers.

Learning outcomes

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

• Explain the properties of the probability spaces one can use for building models for simple experiments.
• Compute the probabilities of complicated events using countable additivity.
• Properly formulate the notion of statistical independence.
• Describe the basic theory of integration, particularly as applied to the expectation of random variables, and be able to compute expectations from first principles.
• Identify convergence in probability and almost sure convergence of sequences of random variables, and use and justify convergence in the computation of integrals and expectations.

Indicative reading list

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: 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.