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# Mathematical statistics

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This online module provides you with the mathematical underpinning for statistical methods in general and – in particular – for other OU statistics modules. You will gain a thorough grounding in mathematical statistics, together with generic skills. You will study distribution theory, leading on to the theory of statistical inference developed under both classical and Bayesian approaches. In the classical case, you will focus on maximum likelihood estimation. You’ll also explore the development of these ideas in the context of linear modelling (regression and extensions). To study this module, you should have a sound knowledge of basic statistical ideas and competence in calculus, algebra and matrices, as provided by the appropriate OU level 1 and 2 study.

## 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.

Read the full content list here.

### 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 IMA website.

## Entry requirements

There is no formal pre-requisite study, but you must have the required mathematical and statistical skills.

## Preparatory work

You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz here, 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

• a week-by-week study planner
• course-specific module materials
• audio and video content
• assessment details, instructions and guidance
• online tutorial access

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.

## You will need

Calculator with basic mathematical functions (exp, log, etc.), but not necessarily with statistical functions.

### Computing requirements

A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

A desktop or laptop computer with either an up-to-date version of Windows or macOS.

The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).

Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

## Teaching and assessment

You'll have a tutor who will help you with the study material and mark and comment on your written work, and who you can ask for advice and guidance. Your tutor can also provide additional assistance with your study skills, especially if you're new to OU study.

Tutorials are designed to aid student success by providing help and guidance with your studies, including hints and tips to improve your understanding. You are encouraged to attend as many as you can –they are an informal way to ask questions and to feel part of a student community.

Tutorials are mainly online but there may be some face to face to tutorials in a limited number of locations, though we cannot guarantee availability close to where you live. A recording of at least one online tutorial will be made available.

Student numbers on the module, and where tutors are based, will affect which tutor may lead a particular tutorial, and what tutorials are offered.

Contact us if you want to know more about study with The Open University before you register.

### Assessment

The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked assignments (TMAs) on paper or online through the eTMA system. You may want to use the eTMA system for some of your assignments but submit on paper for others. This is entirely your choice.

Although your scores on the TMAs will not contribute directly to your final grade, you will need to complete at least four TMAs and score at least 30% on each of them.

## If you have a disability

The OU strives to make all aspects of study accessible to everyone and this Accessibility Statement outlines what studying M347 involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

## Future availability

Mathematical statistics (M347) starts once a year – in October.