# MATH40480 Probability Theory

Probability theory has its roots in games of chance, such as coin tosses or throwing dice. By playing these games, one develops some probabilistic intuition. Such intuition guided the early development of probability theory and allowed for rigorous mathematical statements to be made concerning such games as well as more complex problems involving randomness.

In this course, we will develop the mathematical tools required for the study of randomness. Measure theory and integration play a fundamental role in this development, as does independence and various notions of convergence of random variables. Having covered the basics of probability and measure, we will learn about (and prove in some cases) some of the most essential results in probability theory such as laws of large numbers, Borel-Cantelli lemmas, and central limit theorem. We will also introduce and study conditional expectations, martingales and Brownian motion.

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Curricular information is subject to change

Learning Outcomes:

On the completion of this module the student should be familiar with the fundamental concepts of probability theory. This includes probability measures, random variables, independence, expectation, modes of convergence, laws of large numbers, central limit theorem, conditional expectation, martingales, Brownian motion. The student will develop their ability to deal with abstract concepts and to relate them to concrete examples. The student's ability to realise and critique proofs and arguments will be enhanced.

Indicative Module Content:

Probability and measure; laws of large numbers; central limit theorem; conditional expectation; martingales; introduction to Brownian motion.

Student Effort Hours:
Student Effort Type Hours
Autonomous Student Learning

72

Lectures

36

Total

108

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations
Learning Recommendations:

Students are strongly recommended to revise Introduction to Probability (STAT20110) and Measure Theory & Integration (MATH30360) prior to commencing the course.

Module Requisites and Incompatibles
:
-

Students must have completed MATH30360 Measure Theory and Integration as a pre requisite for this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade

Not yet recorded.

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour