## Course Descriptions

A number of modules on statistics and probability topics are provided by the Department. A brief outline of these modules is provided here; further information is available by following the links to the module home page. Some of the final year modules may be of interest to postgraduate students.

### 1H Modules

#### Core A (Calculus/Probability)

The applications of probability are diverse, occurring in industry, mathematics, science, technology, medicine, social science, agriculture, etc. In this course the theory of probability is developed but always with applications in mind. Among the topics covered are: probability axioms, conditional probability, special distributions, random variables, expectations, generating functions, applications of probability, laws of large numbers, central limit theorems.

### 2H Modules

#### Statistical Concepts II

Anyone who collects information must decide how to draw useful conclusions from it. For example, can an opinion poll involving maybe 1000 people, be trusted to give an accurate picture of everyone else's opinions? Answering this question requires that we combine our knowledge of probability theory with our understanding of how opinion polls are performed. The kind of reasoning that results is called statistical inference.

Other areas of popular debate where statistical problems arise include understanding the effects of food additives, interpreting the results of clinical trials of medical treatments, the reliability of electrical and other products and the incidence of leukemia near nuclear power stations. A knowledge of statistics is essential not only to those who specialise in studying such phenomena but also to anyone who wishes to develop informed opinions about them. The module will introduce some basic ideas of statistical inference and develop solutions to some standard problems. There are two schools of thought about the fundamental principles of statistics, the Bayesians and the frequentists. The module covers both viewpoints but the majority of methods presented will be the more widely used frequentist ones.

Practical computing sessions, using the freely available statistical package R, will be held throughout the year. They serve two purposes: to bring the module closer to the real world of applied statistics and provide additional insight into the lectured material.

### 3H Modules

#### Decision Theory III

Decision theory concerns problems where we have a choice of decision, and the outcome of our decision is uncertain (which describes most problems!). The main topic of the course is statistical decision theory from a Bayesian point of view, and it further includes several introductory presentations of interesting fields of practical decision problems.

The course starts with an introduction to the ideas of decision analysis and the use of decision trees. Quantification of rewards (by means of utilities) and uncertainties (by means of subjective probabilities) are discussed, and these together form the basis of Bayesian statistical decision theory. Formal methods to use data in decision making are presented, including some attention to sequential analysis. Finally, some attention is paid to group decision making, bargaining and game theory.

#### Operations Research III

As its name implies, operations research involves "research on operations", and it is applied to problems that concern how to conduct and coordinate the operations (activities) within an organisation. The nature of the organisation is essentially immaterial, and, in fact, OR has been applied extensively in such diverse areas as manufacturing, transportation, construction, telecommunications, financial planning, health care, the military and public services to name just a few.

This course is an introduction to mathematical models in operations research. Usually, a mathematical model of a practical situation of interest is developed, and analysis of the model is aimed at gaining more insight into the real world. Many problems that occur ask for optimisation of a function f under some constraints. This method is introduced and applied to several problems, e.g. within transportation.

Many situations of interest in OR involve processes with random aspects and we will introduce stochastic processes to model such situations. An interesting area of application, addressed in this course, is inventory theory, where both deterministic and stochastic models will be studied and applied. Further topics will be chosen from: Markov decision processes; integer programming; nonlinear programming; dynamic programming.

#### Statistical Methods III

The course introduces widely used statistical methods, with theoretical issues kept to a minimum. The course, which includes statistical computing, regression, generalised linear models and multivariate analysis, should be of particular interest to those who intend to follow a career in statistics.

#### Stochastic Processes III (B)

A stochastic process is a mathematical model for a system evolving randomly in time. For example, the size of a biological population or the price of a share may vary in an unpredictable manner. These and many other systems in the physical sciences, biology, economics, engineering and computer sciences may best be modeled in a non-deterministic manner. More technically, a stochastic process is a collection of random quantities indexed by a time parameter. Typically, these quantities are not independent, but have their dependency structure specified via the time parameter. Specific models to be covered include Markov chains in discrete and continuous time, Poisson processes and Gaussian processes.

### 4H Modules

#### Bayesian Statistics IV

This course provides an overview and substantial practical applications in Bayesian statistics. The course starts with an introduction to 'the Bayesian approach', which includes discussion of foundational issues. This is followed by Bayesian forecasting, including discussion of the forecast cycle and working towards good understanding of dynamic linear models, also considering important related aspects such as diagnostics and remodelling.

Another main topic is Bayesian computation, in particular Markov Chain Monte Carlo techniques with applications to Bayesian graphical models, exploring conditional independence assumptions. Examples may include spatial smoothing, image processing and medical diagnosis. The course ends with attention to Bayes linear methods. This combines interesting foundational aspects with clear presentations of practical applications through case-studies. Bayes linear methods use expectation as a primitive, which is adjusted in the light of new information.

See our Bayes linear methods home page for more details.

### Other Modules

#### Data Analysis, Modelling and Simulation (DAMS)

This module is a first course in practical data analysis and computer modelling. No prior statistical knowledge or computing knowledge is assumed. The emphasis of the module is upon the understanding of real-life statistical and mathematical problems, and develops the basic concepts and methods by example. Most practical sessions are devoted to computing to apply and illustrate the material presented in lectures.

#### Statistics

The module is designed to be a first statistics course, suitable for all students satisfying the entrance requirements. No prior statistical knowledge is assumed. The emphasis is upon the understanding of real-life statistical problems, and develops the basic concepts and statistical methods by example. Most practical sessions are devoted to computing with a statistical package to apply and illustrate the material presented in lectures.