Aims - This module aims to continue the study of probability and statistics beyond the basic concepts, to provide further study of probability theory, in particular as it relates to multivariate random variables, and to introduce formal concepts and methods in statistical estimation. It is primarily intended for second year students registered on the undergraduate degree programmes offered by the Department of Statistical Science, or jointly with other departments. It also serves as an optional module for students taking the mathematics and statistics stream of the Natural Sciences degree.

Objectives - On successful completion of the module, a student should have an understanding of the properties of joint distributions of random variables and be able to derive these properties and manipulate them in straightforward situations; recognise the chi-squared; t and F distributions of statistics defined in terms of normal variables; be able to apply the ideas of statistical theory to determine estimators and their properties satisfying a range of estimation criteria.

Applications - As with other core modules in probability and statistics, the material in this module has applications in almost every field of quantitative investigation; the module introduces general-purpose techniques that are applicable in principle to a wide range of real-life situations.

Content - Joint probability distributions: joint and conditional distributions and moments; serial expectation; multinomial and multivariate normal distributions. Transformation of random variables: distributions; approximation of moments; order statistics. Moment and probability generating functions: properties; sums of independent random variables; Central Limit Theorem. Relations between standard distributions: chi-squared, t and F distributions. Statistical estimation: bias, mean square error, consistency; method of moments, least squares, maximum likelihood, Cramér-Rao lower bound. Simple examples will be used throughout to motivate and illustrate the topics discussed.

Key texts - Available from ReadingLists@UCL.