Probability 3
| Code | School | Level | Credits | Semesters |
| MATH2109 | Mathematical Sciences | 2 | 10 | Spring UK |
- Code
- MATH2109
- School
- Mathematical Sciences
- Level
- 2
- Credits
- 10
- Semesters
- Spring UK
Summary
This module introduces stochastic processes - i.e. random processes that evolve with time. The focus is on discrete-time Markov chains, which are fundamental to the wider study of stochastic processes. Topics covered include:
- Discrete-time Markov chains:
- Definition and examples, transition probability matrices, Chapman-Kolmogorov equations.
- Class structures, reducibility and irreducibility, recurrence and transience, periodicity.
- Absorption probabilities and stopping times.
- Convergence to equilibrium, ergodicity.
- Reversibility and detailed balance.
- Random walks – definitions, properties, examples, and links to Brownian motion; relationship to Markov chains.
- Branching processes – definitions, properties, and examples; relationship to Markov chains.
- Renewal processes – definitions, properties, and examples; relationship to Markov chains.
- Poisson process – introduction, properties, and examples.
Target Students
Single Honours and Joint Honours students from the School of Mathematical Sciences.
Co-requisites
Modules you must take in the same academic year, or have taken in a previous year, to enrol in this module:
Classes
- One 1-hour lecture
- One 2-hour lecture
Teaching will be through a variety of methods, with the delivery tailored to the material on a week-by-week basis.
Assessment
- 40% Coursework: Summative assessment based on tasks distributed through the semester.
- 60% Exam (2-hour): Written examination – Spring
Assessed by end of spring semester
Educational Aims
The purpose of this module is to provide a thorough grounding in a broad range of techniques required in the analysis of probabilistic models and stochastic processes. Techniques and concepts for the analysis of discrete time Markov chains, random walks, Brownian motion, branching processes, Poisson process, and renewal processes will be explored. Students will acquire knowledge of probabilistic methods for higher level modules in probability and statistics.Learning Outcomes
A student who completes this module successfully will be able to:
L1 – Select and apply suitable mathematical techniques to apply Markov-chain processes to solve routine and novel problems in a variety of contexts.
L2 – Develop and justify a mathematical framework for solving given routine and novel problems, applying correctly probabilistic tools and methods to produce solutions with a high level of accuracy and rigour.
L3 – Summarise complex ideas clearly and concisely to a wide range of audience, drawing connections between different mathematical concepts where appropriate.
L4 – Present conclusions, verbally or in writing, using structured and mathematically rigorous arguments and contextually appropriate language, defending their arguments, results, choices or assumptions against query and criticism.
L5 – Make effective use of a statistical software package to conduct computations and visualise results for a range of audiences.
Conveners
- Dr Wasiur Khuda Bukhsh