Aims

The aim of the module is to provide an introduction to the formal, measure-theoretic foundations of probability, with special attention to conditional expectation.

Learning objectives

At the end of the module you should be able to...

• Understand the concept of a probability space.
• Understand the formal definition o random variables and the construction of distributions of random variables.
• Understand and be able to compute the expected value of a random variable.
• Understand and apply the definition of conditional expectation of a random variable given a σ-field
• Be introduced to the concept of stochastic process and be able to solve basic problems involving Markov chains and martingales.

Syllabus

• Probability spaces and σ-fields.
• Random variables and distributions .
• σ-field generated by a random variable.
• Expectation of a random variable.
• Conditional Expectation of discrete random variables wrt σ-fields
• Introduction to stochastic processes: Markov chains and martingales.

Recommended texts

I don't follow one single textbook, but here are some good books on the subject.

• Chung, K.L. A Course in Probability Theory.
• G. Grimmett and D. Stirzaker. Probability and random processes, Oxford.
• Rosenthal, J.S. (2006). A first look at rigorous probability theory. Second edition. World Scientific.

Teaching

• Spring Term
• 2 lectures per week
• Weekly Seminar

Assessment

Example: One and a half hour closed examination in week 1 Summer Term (90%), Coursework (10%). Note that coursework submitted after the advertised deadlines will be given a mark of zero.

Elective information

A second course on probability  theory.

Please check prerequisites carefully before asking to take this module as an elective.

Prerequisites

• all Taught in: Probability Theory I 
• all Taught in: Analysis I 
• or Introduction to Statistical Theory I