Introductory description

This module develops and motivate techniques which are in everyday use in probability and statistics, and which are essential to a proper understanding of any second- or third-year course in these subjects. It provides the mathematical background for optimization, regression, and best approximation, and develops mathematical thinking.

This module is core for students with their home department in Statistics, optional for students studying Discrete Mathematics. The module may be available to students from other departments as an unusual option if they have taken the necessary prerequisites.

Pre-requisites

• (MA148 Linear Algebra/MA149 Vectors and Matrices) AND (MA140 Mathematical Analysis 2/MA143 Calculus 2) or equivalents.

To be taken together with:

• ST229 Probability for Mathematical Statistics (Statistics Department students),
• Other students take ST232/ST233.

Leads to:

• ST230 Mathematical Statistics (Statistics Department students).

and other Year 2/3 and 4 modules.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

This module provides interdisciplinary coverage of mathematical techniques applying them to advanced probabilistic methods. The module covers:

1. Multivariable integration and its application to multivariate distributions.
2. Differentiation in multidimensions and applications to optimisation problems, including constrained optimisation.
3. The theory of symmetric matrices and application to covariances matrices.
4. Inner product spaces, orthogonalization which leads to linear regression in statistics.

Learning outcomes

By the end of the module, students should be able to:

• analyse multiple integrals and optimisation problems, then select and apply appropriate techniques to solve these problems.
• analyse matrices and vector spaces, then apply spectral theory to determine their structural properties
• analyse and evaluate different strategies to solve mathematical and probabilistic problems, sometimes in multidimensional contexts.
• communicate solutions to problems accurately with structured and coherent arguments.

Indicative reading list

K. F. Riley; M. P. Hobson; S. J. Bence, Mathematical methods for physics and engineering: a comprehensive guide. 2002, Cambridge University Press. ISBN: 0521813727, 0521890675 (Available online and in print in the Main Campus Library.)

Jim Pitman, Probability. 1999. Springer-Verlag New York Inc. ISBN: 9780387979748. (Available online and in print in the Main Campus Library.)

Allan Gut. An intermediate course in probability. 2009. Springer. ISBN: 9781441901620. Available online and in print in the Main Campus Library.)

View reading list on Talis Aspire

Interdisciplinary

This module breaks through domain specificity to delivery an interdisciplinary experience where neither mathematics nor statistics are privileged disciplines and discipline distinctions disappear.

Subject specific skills

Demonstrate facility with advanced mathematical and probabilistic methods.

Demonstrate knowledge of key mathematical and statistical concepts, both explicitly and by applying them to the solution of mathematical problems

Create structured and coherent arguments communicating them in written form

Analyse problems, abstracting their essential information formulating them using appropriate mathematical language to facilitate their solution.

Transferable skills

Problem solving skills: The module requires students to solve problems presenting their conclusions as logical and coherent arguments.

Written communication skills: Students complete written assessments that require precise and unambiguous communication in the manner and style expected in mathematical sciences.

Verbal communication skills: Students are encouraged to discuss and debate formative assessment and lecture material within small-group tutorials sessions. Students can continually discuss specific aspects of the module with the module leader. This is facilitated by statistics staff office hours.
 
Teaming working and working effectively with others: Students are encouraged to discuss and debate formative assessment and lecture material within small-group tutorials sessions.

Professionalism: Students work autonomously by developing and sustain effective approaches to learning, including time management, organisation, flexibility, creativity, collaboratively and intellectual integrity.