The first part of the module begins with a review of probability spaces, random vectors and distribution functions. We then look at transformations of random variables, which reveal deep connections between various well-known probability distributions. Following this, we develop the theory of mathematical expectation and introduce the multivariate normal distribution. We then study moment generating functions and characteristic functions, and use these to establish some classical limit theorems including the law of large numbers and the central limit theorem. The first part of the module concludes with a study of elementary random processes, including the simple random walk, branching processes, Markov chains and martingales. The second part of the module begins with the theory of parameter estimation, followed by a detailed study of statistical hypothesis testing. We then proceed to introduce order statistics, which lead on to the study of non-parametric statistical tests. The second part of the module concludes with an introduction to Bayesian inference.

Knowledge of probability and statistics is useful in many graduate careers. This module gives students a basic knowledge of methods that are 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

• Appreciate the theoretical foundations of probability theory.
• Derive relationships between different probability distributions.
• Prove fundamental results such as the law of large numbers and the central limit theorem.
• Apply the theory of elementary random processes to practical problems.
• Choose and apply appropriate statistical tests in practical problems, and interpret the results.
• Explain the theoretical basis of estimation and hypothesis testing.
• Perform an elementary Bayesian data analysis.

54 - 50 minute lectures

Partial handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 100 hours private study, including the preparation of solutions to homework sheets. Homework sheets are marked and handed back to prepare students for summative assessment.

Skills:

An ability to perform a range of standard statistical tests, and interpret the results of such tests.

An ability to use calculus and other mathematical techniques in the study of probability and statistics.

Transferable Skills:

An appreciation of data analysis, and knowledge of how to select and interpret appropriate methods for statistical analysis.

An understanding of the use of mathematics in statistical modelling, and an assimilation of mathematical theory studied outside this module into a statistical framework.

Formative assessment is a mixture of exercise sheets and homework sheets. The nature of the exercise sheets is determined by the needs of the group, and these differ from year to year.  Feedback is given to students on their submitted work and their overall progress in achieving the learning outcomes of the module.  

Summative assessment is by a single class test and a written examination at the end of the module. The class test is designed also to be a formative assessment, to help students keep track of their progress in understanding the material. The examination gives students the opportunity to demonstrate that they have achieved a good understanding of the learning outcomes.  There is also an opportunity for students to give evidence of 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 five compulsory questions of variable length, of a standard that an average student should be able easily to complete.  Section B has a choice of three from four equally weighted questions requiring 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.
• Random processes.
• Parameter estimation.
• Hypothesis testing.
• Order statistics.
• Non-parametric tests.
• Bayesian inference.

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