Module MATH44320: Advanced Probability

Department: Mathematical Sciences

MATH44320: Advanced Probability

Prerequisites

• EITHER: [Complex Analysis AND Stochastic Processes] OR [Markov Chains AND Analysis]

Corequisites

• None

Excluded Combination of Modules

• None

Aims

• To explore in depth fundamental probabilistic systems in both discrete and continuous settings. To introduce one or two contemporary topics in probability theory and to develop and apply them.

Content

• In Term 1: Order statistics; Coin tossing and trajectories of random walks; Classical limit theorems; Order statistics; Some non-classical limits; Elements of Brownian motion. .
• In Term 2, one or two of the following topics will be announced to run each year: Random graphs and probabilistic combinatorics; Random walks in space; Geometric probability; Random matrix theory; Probability and phase transition; Conformally invariant probability; Interacting particle systems; Random permutations; Random tilings.

Learning Outcomes

• By the end of the module students will:
• be able to solve seen and unseen problems on the given topics.
• Have a knowledge and understanding of this subject demonstrated through an ability to analyse the behaviour of the probabilistic systems explored in the course.
• Reproduce theoretical mathematics concerning probabilistic systems at a level appropriate, including key definitions and theorems.

• In addition students will have enhanced mathematical skills in the following areas: probabilistic intuition.

• Students will have highly specialised skills in the following areas: problem solving, abstract reasoning, modelling, computation.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
• Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
• The end-of-year examination papers assess the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four assignments per term.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University