Module Identifier MA37410  
Academic Year 2001/2002  
Co-ordinator Dr J Lane  
Semester Semester 1  
Other staff Dr J Basterfield  
Pre-Requisite MA20110 , MA26010  
Course delivery Lecture   19 x 1 hour lectures  
  Seminars / Tutorials   3 x 1 hour example classes  
Assessment Exam   2 Hours (written examination)   100%  
  Resit assessment   2 Hours (written examination)   100%  

General description

Probability theory is one of the great achievements of 20th Century mathematics and a thorough grounding in it is necessary for further study of stochastic modelling and mathematical statistics. This module provides that grounding, and proves the limit theorems which provide foundations for so many large-sample statistical methods. Stochastic Processes are processes that deveop in time in a way that is affected by chance, and are used as models for many situations ranging from physics to manpower planning. This module will look at two different sorts of Stochastic Processes, namely Markov chains (whose future development depends only on their present state not their past history) and Branching Processes.


To introduce the most important tools of probability, including conditional expectations and generating functions; to provide students with experience of their use in a variety of problems including proofs of the Central Limit Theorem and Laws of Large Numbers. To introduce students to Branching Processes, to Markov Chains (discrete time, discrete state) and through them to Stochastic Processes in general.

Learning outcomes

On completion of this module, a student should be able to:


1. CONDITIONAL EXPECTATIONS: Revision of joint and conditional distributions; existence of expectations; E(X/Y); E[E(X/Y)] = EX; conditional variance formula; random sums and applications.
2. GENERATING FUNCTIONS: Moment generating function (mgf): basic properties, evaluation of moments, distribution of independent sums. Weak Law of Large Numbers; Central Limit Theorem, applications. Probability generating function: basic properties, relationship to mgf, evaluation of probabilities and moments, random sums.
3. BRANCHING PROCESSES: Definition and introduction. Generating functions for the generation sizes. Extinction probabilities.
4. MARKOV CHAINS: Introduction - the transition matrix. Irreducible classes. Periodicity. Classification of states by their limiting behaviour. Stationary distributions. Hitting probabilities and expected hitting times. An ergodic theorem.

Reading Lists

** Recommended Text
S M Ross. (1998) A First Course in Probability. 5th. Prentice Hall 0138965234
S M Ross. (1997) An Introduction to Probability Models. 6th. Academic Press 0125984707
** Supplementary Text
W Feller. (1968) An Introduction to Probability Theory and its Applications. Vol I.. Wiley 68011708
G R Grimmett & D R Stirzaker. (1992) Probability and Random Processes. 2nd. Oxford 0198536666
H M Taylor & S Karlin. (1994) An Introduction to Stochastic Modelling. revised. Academic Press 0126848858