# ST342 Mathematics of Random Events

###### Lecturer(s): Dr Wei Wu

**Prerequisite(s): **ST115 Introduction to Probability and ST218 Mathematical Statistics Part A (Statistics students) or ST111/2 Probability A&B and ST220 Introduction to Mathematical Statistics (non-Statistics students).

**Leads to:** ST318 Probability Theory.

**Commitment: **3 lectures per week for 10 weeks. This module runs in Term 1.

**Content: **Imagine picking a real number *x *between 0 and 1 "at random" and with perfect accuracy, so that the probability that this number belongs to any interval within [0,1] is equal to the length of the interval. Can we compute the probability of x belonging to any subset to [0,1]?

To answer this question rigorously we need to develop a mathematical framework in which we can model the notion of picking a real number "at random". The mathematics we need, called measure theory, permeates through much of modern mathematics, probability and statistics.

The aim of the module is to provide an introduction to this theory, concentrating on examples and applications. This course would particularly be useful for students willing to learn more about probability theory, analysis, mathematical finance, and theoretical statistics.

**Aims: **To introduce the concepts of measurable spaces, integral with respect to the Lebesgue measure, independence and modes of convergence, and provide a basis for further studies in Probability, Statistics and Applied Mathematics.

**Objectives: **The course will furnish the students with the material and knowledge to:

- Learn how to compute the probabilities of complicated events using countable additivity.
- Understand the proper formulation of the notion of statistical independence.
- Understand the basic theory of integration, particularly as applied to expectation of random variables, and be able to compute expectations from first principles in simple cases.
- Understand and identify convergence in probability and almost sure convergence of sequences of random variables, and use and justify convergence in the computation of integrals and expectations.

**Assessment: **100% by 2-hour examination (January).

**Feedback:** The results of the January examination will be available in week 10 of term 2.

**Books:**

- D. Williams, Probability with Martingales
- P.E. Pfeiffer, Concepts of Probability theory J. Jacob and Ph. Protter, Probability Essentials

**Examination period: **January