Mathematical statistics
This module provides the mathematical underpinning for statistical methods in general and further statistics modules. You’ll study distribution theory, leading to statistical inference theory developed under classical and Bayesian approaches. In the classical case, you’ll focus on maximum likelihood estimation. You’ll also explore the development of these ideas in the context of linear modelling (regression and extensions).
What you will study
Other OU statistics modules focus on hands-on practical applications of statistical techniques and interpretation of data and statistical analyses. This module complements these modules by providing the mathematical theory underlying the methods and concepts, including a treatment of both classical and Bayesian statistics. A considerable amount of mathematics is sometimes required for this development.
This module is delivered online, with integrated use of exercises, animations, audio and video segments. You will also be provided with printed versions of the main units, extra exercises and a handbook.
The module is divided into four blocks of study.
Block 1: Review and distribution theory
The first block comprises a review unit and units introducing distribution theory. The review is mostly of fundamental statistical ideas of the type taught in Analysing data (M248), (see Entry requirements for details); there is also a speedy reminder of important relevant methods in mathematics, including calculus and matrices. Two units in this block introduce the theory of continuous distributions. You will learn, for example, how to evaluate moments of distributions and about other properties of some important univariate distributions. The mathematical structure of multivariate distributions will be explored, with some emphasis on the multivariate normal distribution.
Block 2: Classical inference
The second block is about the classical approach to statistical inference. You will learn how to use calculus to obtain maximum likelihood estimators of parameters. You will also learn about the properties of maximum likelihood estimation and of point estimation more generally. The mathematics underlying hypothesis tests and confidence intervals will be explored. There is also a unit on asymptotic (large sample) analysis, giving an insight into how statisticians study properties of statistical procedures by approximate methods.
Block 3: Bayesian statistics
In the third block you’ll consider the Bayesian approach to statistical inference. The emphasis is first on so-called conjugate analysis which constitutes the type of Bayesian analysis most amenable to straightforward mathematical development. You’ll consider prior to posterior analysis first, followed by Bayesian estimation based on decision theory. Markov chain Monte Carlo (MCMC) is a technique often used for tackling Bayesian problems which are not conjugate; you’ll investigate the mathematical ideas leading to the basic methods of MCMC.
Block 4: Linear modelling
The fourth and final block gives some of the mathematical development underlying linear modelling. The material covers linear regression on a single explanatory variable; multiple linear regression where there is more than one explanatory variable; and generalised linear modelling for regression situations where the normal distribution is not a suitable model for variation in the response. Both classical and Bayesian approaches to the analysis of these models are considered.
You can find the full content list on the .
You will learn
Successful study of this module should enhance your skills in understanding some useful mathematical theory, interpreting mathematical results in a statistical context, constructing logical arguments, and finding solutions to problems.
Professional recognition
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the .
This module may also help you to apply for the professional award of Graduate Statistician conferred by .
Entry requirements
There is no formal pre-requisite study, but you must have the required mathematical and statistical skills.
You can .
if you’re not sure if you’re ready.
Preparatory work
You should aim to be confident and fluent with the concepts covered in the , and follow the advice in the quiz.
The key topics to revise include:
- matrices
- basic differentiation and integration.
You’ll also find it useful to be familiar with the following topics:
- normal, Poisson and binomial distributions
- the central limit theorem
- point estimation
- maximum likelihood estimation
- confidence intervals
- hypothesis testing
- simple linear regression
- correlation.
An OU level 2 module in mathematics is ideal preparation, and Analysing data (M248) is also useful.
What's included
You’ll have access to a module website, which includes:
- a week-by-week study planner
- course-specific module materials
- audio and video content
- assessment details, instructions and guidance
- online tutorial access
- access to student and tutor group forums.
You’ll be provided with printed books covering the content of the module, including explanations, examples and activities to aid your understanding of the concepts and associated skills and techniques. You’ll also receive a printed module handbook.
Computing requirements
You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.
Any additional software will be provided or is generally freely available.
To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).
Our module websites comply with web standards, and any modern browser is suitable for most activities.
Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.
It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.