# 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 statements to be made concerning such games as well as more complex situations such as the evolution of stock prices.

In this course, we will develop the mathematical tools required to study sequences of real-valued random variables. We will use these tools to study some of the most central objects in probability. We will then see how these can be used to model real-world applications such as betting strategies.

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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 which lead towards the deep mathematical theory of Stochastic Analysis. This includes independence, expectation, conditional expectation, stochastic processes, filtrations, martingales.
The student will develop their ability to deal with abstract concepts and to relate them to real world examples. The student's ability to realise and critique proofs and arguments will be enhanced.

Indicative Module Content:

Measure theoretic approach to probability theory; types of convergence for random sequences; independence of sigma algebras; Borel-Cantelli lemmas; laws of large numbers; conditional probability and expectation; martingale convergence theorems; optional stopping theorems for martingales. Time permitting, central limit theorems and/or Brownian motion.

Student Effort Hours:
Student Effort Type Hours
Lectures

36

Autonomous Student Learning

72

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
Continuous Assessment: Homework sheets. Throughout the Trimester n/a Standard conversion grade scale 40% No

20

Examination: 2 hour exam. 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

80

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour