## MAS275 Probability Modelling

Note: This is an old module occurrence.

You may wish to visit the module list for information on current teaching.

 Semester 2, 2019/20 10 Credits Lecturer: Dr Robin Nicholas Stephenson Home page 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.

Discrete time renewal processes
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
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.