Module Identifier MA37410  
Academic Year 2000/2001  
Co-ordinator Dr J A Lane  
Semester Semester 2  
Pre-Requisite MA20110 , MA26010  
Course delivery Lecture   19 x 1hour lectures  
  Seminars / Tutorials   3 x 1hour 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 staistics. 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 in 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. A First Course in Probability. Prentice Hall
S M Ross. An Introduction to Probability Models. Academic Press
** Supplementary Text
W Feller. An Introduction to Probability Theory and its Applications. Wiley
G R Grimmett & D R Stirzaker. Probability and Random Processes. Oxford
H M Taylor & S Karlin. An Introduction to Stochastic Modelling. revised. Academic Press