Module Identifier MA37410  
Academic Year 2003/2004  
Co-ordinator Dr John A Lane  
Semester Semester 2  
Other staff Dr John A Lane, Dr J G Basterfield  
Pre-Requisite MA20110 , MA26010  
Course delivery Lecture   19 x 1 hour lectures  
  Seminars / Tutorials   3 x 1 hour example classes  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

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

Brief 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 develop 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 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 the Weak Law of Large Numbers. To introduce students to Branching Processes, to Markov Chains (discrete time, discrete state), and through them to Stochastic Processes in general.


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


This module is at CQFW Level 6