# Module Details

## MA922-7-AU-CO: Stochastic Processes

Note: This module is inactive. Visit the Module Directory to view modules and variants offered during the current academic year.

Year: 2017/18
Department: Mathematical Sciences
Essex credit: 15
ECTS credit: 7.5
Available to Study Abroad / Exchange Students: Yes
Full Year Module Available to Study Abroad / Exchange Students for a Single Term: No
Outside Option: No

Staff
Supervisor:
Teaching Staff: Dr. David Penman, email: dbpenman@essex.ac.uk
Contact details: Miss Shauna McNally - Graduate Administrator. email: smcnally@essex.ac.uk, Tel 01206 872704

 Module is taught during the following terms Autumn Spring Summer

### Module Description

Aims
To acquaint students with the use of probability theory to study models of phenomena with a degree of unpredicability about them, such as queues and population growth.

Syllabus

- Examples of stochastic processes.
-Markov chains: definition and examples; random walk, gambler's ruin.
- Chapman-Kolmogorov equation.
- Classification of states: persistent (null or non-null) or transient, periodic or aperidoic.
- Probability generating functions.
- Limiting probabilities.
- Absorption probabilities.
- Branching processes: number of individuals in nth generation. Probability of extinction. Total population until extinction.
- Exponential distribution: definition and properties, memoryless property.
- Continuous-time Markov chains: definition. Birth and death processes.
- Chapman-Kolmogorov and Kolmogorov equations. Limiting probabilities. Balance equations.

On completion of the module, students should be able to:
- understand the Markov property and its application to Markov chains in both discrete and continuous time;
- derive the Chapman-Kolmogorov equation in both discrete and continuous time;
- classify the states of Markov chains;
- calculate mean recurrence times, limiting probability distributions and absorption probabilities for discrete-time Markov chains;
- calculate population-size distributions and extinction probabilities for discrete-time branching processes;
- understand the memoryless property of the exponential distribution;
- derive the Kolmogorov forward and backward equations and use them to find transition probabilities of continuous-time Markov chains.
use balance equations to find limiting probabilities of continuous-time Markov chains.

### Learning and Teaching Methods

The module runs at 3 hours per week. There are 5 lectures and one class in every fortnight.

In the Summer term 3 revision lectures are given.

### Assessment

20 per cent Coursework Mark, 80 per cent Exam Mark

### Other details

Information about coursework deadlines can be found in the "Coursework Information" section of the Current Students, Useful Information Maths web pages: Coursework and Test Information

### Exam Duration and Period

2:00 during Summer Examination period.