# Foundations of Probability

## Content

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## Description

#### Module Overview

Probability theory provides the mathematical foundations for both statistical inference and stochastic modelling, e.g. of physical, biological and environmental processes.

The subject is very extensive. Treatments range from elementary undergraduate level (coin tossing, etc) to highly sophisticated accounts based on extensive use of measure theory and functional analysis. They also vary from those which concentrate almost exclusively on the mathematical foundations to those focused on the modelling of applications.

The aim of the present module is to provide a concise introduction to probability theory, developing both the careful mathematical reasoning which that subject requires and also the physical insight into the behaviour of random phenomena which motivates and guides the subject. It is the almost unique interplay between these two aspects of the discipline which make it so interesting.

#### Module Overview

Foundations of Probability (semester 1)

The first seven lectures are concerned with the mathematical basis of probability. There is a mathematically careful development of material which many will be familiar with from more elementary courses: probability spaces, conditioning, independence, random variables, distributions, convergence concepts and laws of large numbers. With regard to the measure-theoretic foundations of the subject we have aimed to strike a careful balance: there is a necessary treatment of the concept of measurability, and in particular of the central role played of sigma-algebras throughout probability, notably in the consideration of conditioning and independence.

The next two lectures are concerned with conditional expectation and martingales. The final lecture looks at elements of renewal theory. These topics play a crucial role in understanding the behaviour of random processes evolving in time, which will be looked at further in the second semester module Stochastic Processes.

##### Lecturers:

- Michela Ottobre (Heriot-Watt University)
- Burak Buke (University of Edinburgh)
- Sergey Foss (Heriot-Watt University)
- Fraser Daly (Heriot-Watt University)

**Assessment**

This module is assessed by two written assignments (to be set at least two weeks before the deadline), to be submitted, provisionally, by mid November 2020 (Assignment 1) and early January 2021 (Assignment 2). Exact dates will be confirmed nearer the time. Solutions to at least one of the assignments should be produced using LaTeX.

##### Prerequisites

Elements of mathematical analysis, linear algebra and combinatorics at undergraduate level.