Modelling events in time

Introduction

Please note: a Statement of Participation is not issued on this course.

This free OpenLearn course, Modelling events in time, is an extract from the Open University course M343 Applications of probability [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a third level course that develops ideas about probability and random processes that were introduced in the Open University course M248 Analysing data. The approach adopted is to develop a probability model for a practical situation and then to investigate the properties of the model. Students are expected to be familiar with basic probability, random variables and expectation and the Poisson process.

Modelling events in time consists of material from M343 Book 2, Modelling events in time and space, and has six sections in total. You should set aside about two to three hours to study each of the sections; the whole extract should take about 16 hours to study. The extract is a small part (around 8%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

Many events occur in a random, unpredictable way and this extract focuses on modelling the patterns formed by events occurring in time. It is relatively self-contained and should be reasonably easy to understand for someone with a good knowledge of statistics, such as could be gained from studying the Open University course M248 Analysing data. A few techniques and definitions are present in the extract without explanation.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M343 software or to video material (although please note that the PDF may contain references to other parts of M343). In this extract, some illustrations have also been removed due to copyright restrictions.

Regrettably, mathematical and statistical content in PDF form is not accessible using a screenreader, and you may need additional help to read these documents.

Sections 1 and 2 introduce the fundamental ideas of random processes through a series of examples. The notation used for random processes is introduced in Section 1 and some further examples are described in Section 2.

Section 3 describes a model that is appropriate for events occurring ‘at random’ in such a way that their rate of occurrence remains constant. This model is known as the Poisson process.

Section 4 derives the main results, quoted without proof in Section 3, using a mathematical approach.

Section 5 introduces the multivariate Poisson process in which each event may be just one of several different types of event.

Section 6 introduces the non-homogeneous Poisson process in which events occur at a rate that varies with time.