Unit name | Further Topics In Probability 4 |
---|---|

Unit code | MATHM0018 |

Credit points | 20 |

Level of study | M/7 |

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

Unit director | Dr. Balazs |

Open unit status | Not open |

Pre-requisites |
Applied Probability 2 (MATH 21400). |

Co-requisites |
None. |

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.

**General Description of the Unit**

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.

**Relation to Other Units**

Measure Theory & Integration (MATH34000) & Metric Spaces (MATH20200) are recommended.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

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.

Lectures supported by problem sheets and solution sheets.

80% Examination and 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.

- R. Durrett: Probability – Theory and Examples, Duxbury Press, 1995

- W. Feller: An introduction to probability theory and its applications. Vols.1, 2. Wiley, 1970

- J. Lamperti: Probability -- a Survey of the Mathematical Theory , W.A Benjamin Inc., New York-Amsterdam, 1966

- S. Resnick: Adventures in Stochastic Processes, Birkhauser, 1992

- A. N. Shiryaev: Probability (Second Edition), Springer, Graduate Texts in Mathematics 95, 1996

- D. Williams: Probability with martingales. Cambridge UP, 1990

Instructor’s lecture notes and problem sheets