Undergraduate Programme and Module Handbook 2024-2025

# Module MATH2707: Markov Chains II

## Department: Mathematical Sciences

### MATH2707: Markov Chains II

Type | Open | Level | 2 | Credits | 10 | Availability | Available in 2024/2025 | Module Cap | None. | Location | Durham |
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#### 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 Combination 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

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 21 | 2 per week in Michaelmas; 1 in Easter | 1 hour | 21 | |

Tutorials | 5 | Fortnightly in Michaelmas; 1 in Easter | 1 hour | 5 | ■ |

Problem Classes | 4 | Fortnightly in Michaelmas | 1 hour | 4 | |

Preparation and reading | 70 | ||||

Total | 100 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written Examination | 2 hours | 100% |

#### Formative Assessment:

Weekly written or electronic assignments to be assessed and returned.

■ 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