The first part of the module begins with the study of probability spaces, random variables and distribution functions. We then look at various transformations of random variables, which reveals deep connections between many well-known probability distributions. Following this, we develop the theory of mathematical expectation and conditional expectation. The first part of the module concludes with the study of moment generating functions and characteristic functions, which are then 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, along with an introduction to Bayesian inference. We then develop the theory of statistical hypothesis testing, focusing in particular on the likelihood ratio test. We then introduce order statistics, and proceed to study a number of different non-parametric tests and their various applications. The second part of the module concludes with a brief look at linear models, including least squares and logistic regression.

Knowledge of probability and statistics is useful in many graduate careers. This 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.

Prerequisite Modules: MA1500 Introduction to Probability Theory, MA1501 Statistical Inference

• Understand the theoretical foundations of probability and statistics.
• Derive relationships between different probability distributions.
• Prove fundamental results such as the law of large numbers and the central limit theorem.
• Explain the theoretical basis of parameter estimation and hypothesis testing.
• Choose and apply appropriate statistical tests in practical problems, and interpret their results.
• Use statistical techniques based on the general linear model.

54 - 50 minute lectures

Students are expected to take notes during lectures. Copies of the slides 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 a series of homework assignments, and a written examination at the end of the module. The homework assignments are also intended as formative assessments, to help students keep track of their progress in understanding the material. 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.
• Transformations of random variables.
• Mathematical expectation.
• Generating functions.
• Limit theorems.
• Estimation theory.
• Hypothesis testing.
• Non-parametric methods.
• Linear models.

Not applicable.

Probability and Random Processes (3rd ed), Grimmett, G. R. & Stirzaker, D. R., Oxford University Press, 2001

Introduction to Mathematical Statistics (6th ed), Hogg, R.V.,  McKean, J. W., & Craig, A.T., Prentice Hall, 2005