Knowledge of probability and statistics is useful in many graduate careers. This double module gives students an understanding of the principles underlying statistical methods commonly used by professional statisticians, and is intended to prepare students for a career involving statistical analysis.

The first part of the module begins with the study of probability spaces, random variables and distributions, followed by the theory of mathematical expectation and conditional expectation. We then look at moment generating functions, which are used to prove classical limit theorems such as the law of large numbers and the central limit theorem. The second part of the module begins with a study of parameter estimation, including the notions of consistency and efficiency, and an introduction to Bayesian inference. We then look at the theory of statistical hypothesis testing, focusing in particular on the likelihood ratio test and a number of different non-parametric tests.

Prerequisite Module: MA1500 Introduction to Probability Theory

Recommended Module: MA1501 Statistical Inference

• Understand the theoretical foundations of probability and statistics.
• Derive relationships between different probability distributions.
• Prove fundamental results such as the central limit theorem.
• Explain the theoretical basis of parameter estimation and hypothesis testing

44 fifty-minute lectures

11 fifty-minute exercise classes

Students are expected to take notes during lectures. Handouts will be available on Learning Central. Students are also expected to undertake at least 100 hours private study, involving regularly reviewing lecture notes, engagement with exercise sheets, and preparing homework submissions.

Skills:

An ability to apply various mathematical ideas and techniques to the study of probability and statistics.

An ability to choose appropriate statistical tests in different contexts, and to perform and interpret the results of such tests.

Transferable Skills:

Formulating problems and presenting clear solutions through logical reasoning.

Formative assessment is by a series of exercise sheets. Feedback is given to students on their submitted work, and on their overall progress in achieving the learning outcomes of the module.  

Summative assessment is by written examination at the end of the module.  The examination gives students the opportunity to demonstrate that they have achieved the learning outcomes of the module.  There is also an opportunity for students show a depth of understanding that merits the award of higher than average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions, of a standard that an average student should be able to complete comfortably. Section B has a choice of three from four equally weighted questions that require a greater depth of understanding than those in Section A.

• Probability spaces.
• Random variables and distributions.
• Expectation.
• Joint distributions.
• Sums of random variables
• Estimation.
• Hypothesis testing.

Grimmett, G.R. and Stirzaker, D.R. 2001. Probability and random processes. 3rd ed. Oxford University Press.

Hogg, R.V. et al. 2013. Introduction to mathematical statistics. 7th ed. Prentice Hall.