# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 18 Hours. (18 x 1 hour lectures) |

Seminars / Tutorials | 4 Hours. (4 x 1 hour example classes) |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours (written examination) | 100% |

Supplementary Exam | 2 Hours (written examination) | 100% |

### Learning Outcomes

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

1. use conditional expectations and variances in a wide range of problems involving stochastic modelling;

2. calculate generating functions of standard univariate distributions and use them to calculate moments and distributions of sums and mixtures;

3. use generating functions in limiting arguments, including the proofs of the Central Limit Theorem and the Weak Law of Large Numbers;

4. explain the need for careful mathematical argument in probability theory;

5. describe and use the recurrence relation for generation sizes in a Branching Process and determine the probability of ultimate extinction;

6. determine the structure of the state space of a Markov chain from its transition matrix;

7. classify the states of Markov chain by period and limiting behaviour, including the calculation of appropriate limiting probabilities;

8. explain the Ergodic Theorem for Markov chains and use it to calculate limiting time averages from limiting probabilities.

### 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.

### Aims

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.

### Content

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 List

**General Text**

Jones, P. W. (2009.) Stochastic processes :an introduction /P.W. Jones and Peter Smith. Chapman &amp; Hall Primo search Pinsky, Mark A. (2011.) An introduction to stochastic modeling /Mark A. Pinsky, Samuel Karlin. 4th ed. Academic Press Primo search

**Recommended Text**

Ghahramani, Saeed. (c2005.) Fundamentals of probability with stochastic processes /Saeed Ghahramani. Pearson/Prentice Hall Primo search S M Ross (1998) A First Course in Probability 5th Prentice Hall Primo search S M Ross (1997) An Introduction to Probability Models 6th Academic Press Primo search Stirzaker, David. (2005.) Stochastic processes and models /David Stirzaker. Oxford University Press Primo search Weiss, N. A. (c2006.) A course in probability /Neil A. Weiss, with contributions from Paul T. Holmes, Michael Hardy. Pearson Addison Wesley Primo search

**Supplementary Text**

Grimmett, Geoffrey. (2001 (2006 prin) Probability and random processes /Geoffrey R. Grimmett and David R. Stirzaker. Oxford University Press Primo search H M Taylor & S Karlin (1994) An Introduction to Stochastic Modelling revised Academic Press Primo search W Feller (1968) An Introduction to Probability Theory and its Applications, Vol I Wiley Primo search

### Notes

This module is at CQFW Level 6