Aims

To develop ideas and methods that are essential for the study of probability and statistics. To develop concepts in probability that underpin methods of statistical inference.

By the completion of the course, students will be familiar with ideas of statistical modelling, data analysis and interpretation.

Module summary

The course will cover the key concepts required for further study of probability and statistics. We begin with the fundamentals of probability theory, considering probability for discrete outcomes such as National Lottery draws or poker hands. We will then move on to probability distributions and investigate how they can be used to model uncertain quantities such as the response of patients to a new treatment in a clinical trial and the occurrence of earthquakes in tectonically active regions.

Next we will look at using observed samples of data to estimate properties of a population.
For example, we will see how we can use information on the actual survival rates of a group of patients to make statements about survival rates of patients in general. Probability theory will be used to establish how confident we can be about the accuracy of such statements and key statistical ideas will be introduced in the examination of these questions.

Outline Of Syllabus

Introduction to random variation and probability: probability axioms. Counting arguments. Conditional probability and independence. Discrete probability models: the binomial, geometric and Poisson distributions. Continuous probability models: the uniform. exponential and Normal distributions. Calculation and interpretation of mean and variance. Introduction to Statistical Inference: estimation of population quantities, properties of estimators. Law of large numbers and central limit theorem. Method of moments. Introduction to likelihood and Bayesian inference: maximum likelihood, prior and posterior distributions.

Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of the programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one way for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.