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MATH2707: Markov Chains II

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Type Open
Level 2
Credits 10
Availability Available in 2024/2025
Module Cap None.
Location Durham
Department Mathematical Sciences

Prerequisites

  • [Calculus I (Maths Hons) (MATH1081) OR Calculus I (MATH1061)] AND [Linear Algebra I (Maths Hons) (MATH1091) OR Linear Algebra I (MATH1071)] AND Probability I (MATH1597) AND Analysis I (MATH1051) [the latter may be a co-requisite].

Corequisites

  • Analysis I (MATH1051) unless taken before.

Excluded Combinations of Modules

  • None

Aims

  • To introduce and develop the concept of a Markov chain, as a fundamental type of stochastic process, and to study key features of Markov models using probabilistic tools such as generating functions.

Content

  • Markov property
  • Stationary distributions
  • Classification of states
  • Hitting probabilities and expected hitting times
  • Convergence to equilibrium
  • Applications to random walks
  • Generating functions
  • Further topics to be chosen from: Gibbs sampler, mixing and card shuffling, stochastic epidemics, discrete renewal theory, Markov decision processes

Learning Outcomes

Subject-specific Knowledge:

  • By the end of this module students will:
  • Be able to solve seen and unseen problems involving Markov chains.
  • Have a knowledge and understanding of this subject demonstrated through an ability to identify stationary distributions, classify states, and compute hitting probabilities and expected hitting times, and a working knowledge of generating functions and their computational and theoretical power.
  • Reproduce theoretical mathematics concerning Markov chains to a level appropriate to Level 2, including key definitions and theorems.

Subject-specific Skills:

  • Students will have enhanced mathematical skills in the following areas: intuition for key features of probabilistic systems that evolve in time.

Key Skills:

  • Students will have basic mathematical skills in the following areas: problem solving, 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.
  • Tutorials provide active problem-solving engagement and immediate feedback to the learning process.
  • 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 assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures212 per week in Michaelmas; 1 in Easter1 hour21 
Tutorials5Fortnightly in Michaelmas; 1 in Easter1 hour5Yes
Problem Classes4Fortnightly in Michaelmas1 hour4 
Preparation and reading70 
Total100 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
Written Examination2 hours100 

Formative Assessment

Weekly written or electronic assignments to be assessed and returned.

More information

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