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:

MAS352Stochastic Processes and Finance
MAS371Applied Probability
MAS452Stochastic 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