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.

Aims
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:

• use conditional expectations and variances in a wide range of problems involving stochastic modelling;
• calculate generating functions of standard univariate distributions and use them to calculate moments and distributions of sums and mixtures;
• use generating functions in limiting arguments, including the proofs of the Central Limit Theorem and the Weak Law of Large Numbers;
• explain the need for careful mathematival argument in probability theory;
• describe and use the recurrence relation for generation sizes in a Branching Process and determine the probability of ultimate extinction;
• determine the structure of the state space of a Markov chain from its transition matrix;
• classify the states of Markov chain by period and limiting behaviour, including the calculation of appropriate limiting probabilities;
• explain the Ergodic Theorem for Markov chains and use it to calculate limiting time averages from limiting probabilities.

Syllabus
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.