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MATH44320: Advanced Probability

It is possible that changes to modules or programmes might need to be made during the academic year, in response to the impact of Covid-19 and/or any further changes in public health advice.

Type Tied
Level 4
Credits 20
Availability Available in 2024/2025
Module Cap None.
Location Durham
Department Mathematical Sciences

Prerequisites

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

Corequisites

  • None

Excluded Combinations 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

Subject-specific Knowledge:

  • 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.

Subject-specific Skills:

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

Key Skills:

  • 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

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week in Michaelmas term; 2 per week in Epiphay term; 2 in week 211 hour42 
Problems classes8Fortnightly in Michaelmas and Epiphany1 hour8 
Preparation & reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
End of year written examination 3 hours100 

Formative Assessment

Four assignments per term.

More information

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