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# Univariate continuous distribution theory

## Introduction

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This free OpenLearn course, Univariate continuous distribution theory, is an extract from the Open University course M347 Mathematical statistics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a third level course that provides the mathematical theory underlying the methods and concepts used in practical statistical analyses. Students are expected to have a basic knowledge of the ideas and concepts of statistical science. A considerable amount of mathematics is also sometimes required for the development of theory presented in the course.

Univariate continuous distribution theory consists of material from M347 Unit 2, Univariate continuous distribution theory, and has two sections in total. The whole extract should take about 10 hours to study, and since section 2 is longer than section 1, you should set aside more time to study section 2. The extract is a small part (around 4%) 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.

This extract looks at a number of the basic properties of statistical models. You should already be familiar with many of the ideas presented, however, some mathematical flesh will be added to the conceptual and data-based knowledge that you have gained from less mathematical courses of study. 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. The mathematical knowledge required could be gained from studying up to the level of the Open University course MST125 Essential mathematics 2 (or its predecessor MS221 Exploring mathematics). The key mathematical technique is integration and inevitably there will be some algebraic manipulation. 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 M347 software or to video material (although please note that the PDF may contain references to other parts of M347). 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.

Section 1 is concerned with the distributions of continuous random variables which are described by their probability density functions (pdfs) and cumulative distribution functions (cdfs). The basic properties of pdfs are considered in Subsection 1.1 and the basic properties of cdfs are considered in Subsection 1.2. To end the first section, you will be reminded how differences between cdf values give probabilities of a random variable lying within a given interval.

Section 2 is concerned with moments. The concept of expectation or expected value is described in Subsection 2.1. The familiar notion of the mean, also known as the first moment, is then considered in Subsection 2.2. Subsections 2.3 and 2.4 cover two general definitions of moments. Then variance is covered in Subsection 2.5. Subsection 2.6 concerns random variables linked by linear transformation and finally in Subsection 2.7, you will see how to deal with moments all in one go using the moment generating function.