Unit name | Further Topics In Probability 3 |
---|---|

Unit code | MATH30006 |

Credit points | 20 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Dr. Balazs |

Open unit status | Not open |

Pre-requisites |
MATH20008 Probability 2 |

Co-requisites |
MATH34000 Measure Theory and Integration would be useful but is not essential |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Unit Aims**

To outline, discuss, and prove with full mathematical rigour some of the key results in classical probability theory; with special emphasis on applications.

**Unit Description**

This course deals with various analytic tools used and exploited in probability theory. Various modes of convergence of random variables (almost surely, weak, in probability, in Lp and in distribution) and the connections between them are presented. The key theorems are the Weak and Strong Laws of Large Numbers and the Central Limit Theorem. The analytic tools are: generating functions, Laplace- and Fourier transforms and fine analysis thereof.

Learning Objectives

To gain profound understanding of the basic notions and techniques of analytic methods in probability theory. In particular: generating functions, Laplace- and Fourier-transforms. To gain insight and familiarity with the various notions of convergence in the theory of random variables (in probability, almost sure, L^p, in distribution). Special emphasis will be on various “down-to-earth” applications of the mathematical theory.

The unit will be taught through a combination of

- synchronous online and, if subsequently possible, face-to-face lectures
- asynchronous online materials, including narrated presentations and worked examples
- guided asynchronous independent activities such as problem sheets and/or other exercises
- synchronous weekly group problem/example classes, workshops and/or tutorials
- synchronous weekly group tutorials
- synchronous weekly office hours

80% Timed, open-book examination 20% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

**Recommended**

- Richard Durrett,
*Probability – Theory and Examples*, Duxbury Press, 1995 - William Feller,
*An Introduction to Probability Theory and its Applications*. Vols.1, 2, Wiley, 1970 - John Lamperti,
*Probability - A Survey of the Mathematical Theory*, W.A Benjamin Inc., New York-Amsterdam, 1966 - Sidney Resnick,
*Adventures in Stochastic Processes*, Birkhauser, 1992 - Albert N. Shiryaev,
*Probability (Second Edition)*, Springer, Graduate Texts in Mathematics 95, 1996 - David Williams,
*Probability with Martingales*, Cambridge University Press, 1991