Aims

To develop skills in probabilistic reasoning and to gain familiarity with some of the main techniques involved in the analysis of stochastic processes. To lay foundations for further study of probability and statistics.

Module Summary
How can we model the size of a population as it changes over time? How can we predict the chance of a gambler winning given a certain strategy? On average how much will they win? Undoubtedly, significant uncertainty is a central feature of almost every real-life problem, and questions of this nature arise naturally in many applications ranging from economics and finance through to engineering, microbiology and genetics. It is precisely this element of uncertainty which makes questions like “How should I invest in the stock market?” challenging and interesting. In this course, you will discover how certain probabilistic techniques can be used to model and analyse systems or phenomena that evolve randomly over time.

Outline Of Syllabus

Review of probability ideas: conditioning arguments; Random walks; definition of Markov chains & the Markov property; Transition probabilities; Chapman-Kolmogorov equations; Classification of states, communicating classes; Invariant and reversible distributions; Branching processes in discrete time; Queues.

Teaching Rationale And Relationship

Teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking. Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem classes are used to help develop the students’ abilities at applying the theory to solving problems. Module drop-in sessions allow students to receive learning support in areas where they may need additional guidance.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Exam problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.