## MAS275 Probability Modelling

 Semester 2, 2013/14 10 Credits Lecturer: Dr Jonathan Jordan 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: MAS201 (Linear Mathematics for Applications); MAS202 (Advanced Calculus); MAS205 (Statistics Core)

The following modules have this module as a prerequisite:

 MAS352 Stochastic Processes and Finance MAS371 Applied Probability MAS452 Stochastic Processes and Finance

## Outline syllabus

• Discrete time renewal processes
• Discrete time Markov chains
• Random point processes in time and space

## 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.

21 lectures, 5 tutorials

## Assessment

One formal 2 hour closed book examination.

## Full syllabus

• Discrete time renewal processes: Definition, generating functions, the renewal theorem, delayed renewal processes.
• Discrete time Markov chains: Transition probabilities, classification of states, equilibrium and absorption probabilities. Examples: gambler's ruin, inventory models, dam models, diffusion of particles.
• Random point processes in time and space: Poisson process: superposition, censoring, conditioning on number of events in an interval. Inhomogeneous, compound 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 519.2 (F) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.

## Timetable

 Mon 11:00 - 11:50 lecture Arts Tower Lecture Theatre 3 Tue 12:00 - 12:50 tutorial (group 81) (even weeks) K14 Hicks Building Wed 11:00 - 11:50 lecture Psychology Lecture Theatre G30 Wed 12:00 - 12:50 tutorial (group 82) (even weeks) Hicks Lecture Theatre 4 Thu 12:00 - 12:50 tutorial (group 83) (even weeks) Hicks Lecture Theatre 10