MAS275 Probability Modelling
| Semester 2, 2021/22 | 10 Credits | ||||
| Lecturer: | Dr Robin Nicholas Stephenson | Home page | Timetable | Reading List | |
| Aims | Outcomes | Assessment | Full Syllabus | ||
The course introduces a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might be the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. The aim is to familiarize students with an important area of probability modelling.
Prerequisites: MAS113 (Introduction to Probability and Statistics); MAS211 (Advanced Calculus and Linear Algebra)
The following modules have this module as a prerequisite:
| MAS352 | Stochastic Processes and Finance |
| MAS371 | Applied Probability |
| MAS452 | Stochastic Processes and Finance |
Outline syllabus
- Introduction to Markov chains
- Discrete time renewal theory
- Limiting behaviour of Markov chains
- Applications of Markov chains
- Hitting times and probabilities
- Poisson processes
Aims
- To introduce and study a number of general models for processes where the state of a system is fluctuating over a period of time according to some random mechanism.
- To illustrate the above models by example and by simulation.
- To familiarise students with an important area of probability modelling.
Learning outcomes
- model a range of situations by writing down transition matrices for suitable Markov chains.
- calculate and interpret equilibrium probabilities and distributions of Markov chains.
- calculate and interpret absorption probabilities and expected times to absorption in Markov chains.
- understand the special properties of the simple Poisson process, perform calculations with them and interpret the results.
- understand the spatial and inhomogeneous extensions of the Poisson process, and apply them as models of real phenomena.
22 lectures, 5 tutorials
Assessment
One formal 2 hour closed book examination.
Full syllabus
Introduction to Markov chains
Definition, transition probabilities, examples including random walks and gambler's ruin, Chapman-Kolmogorov, stationary distributions.
Definition, generating functions, delayed renewal processes. Limiting behaviour of Markov chains
Classification of states, link to renewal theory, limit theorem, periodic and non-irreducible chains, the renewal theorem. Applications of Markov chains
Google PageRank. Hitting times and probabilities
Hitting probabilities, expected hitting times. Poisson processes
Poisson process: superposition, thinning, conditioning on number of events in an interval. Inhomogeneous and spatial generalisations.
Reading list
| Type | Author(s) | Title | Library | Blackwells | Amazon |
|---|---|---|---|---|---|
| C | E. Parzen | Stochastic Processes | 519.23 (P) | Blackwells | Amazon |
| C | G.R. Grimmett, D.R.Stirzaker | Probability and Random Processes | 519.2 (G) | Blackwells | Amazon |
| C | S.M. Ross | Introduction to Probability Models | 519.2 (R) | Blackwells | Amazon |
| C | W. Feller | An Introduction to Probability Theory and its Applications |
(A = essential, B = recommended, C = background.)
Most books on reading lists should also be available from the Blackwells shop at Jessop West.