ST22810 Mathematical Methods for Statistics and Probability
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 thirdyear 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.
This module has prerequisites: (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), ST232/ST233 (Other students).
Leads To: ST230 Mathematical Statistics (Statistics Department students).
Module aims
The module’s aims are:

Develop students’ mathematical knowledge in multivariable differentiation and integration, matrix theory and linear algebra.

Apply this mathematical knowledge to build advanced probabilistic methods used in statistics such as multivariate probability distributions and conditional distributions.
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:
 Multivariable integration and its application to multivariate distributions.
 Differentiation in multidimensions and applications to optimisation problems, including constrained optimisation.
 The theory of symmetric matrices and application to covariances matrices.
 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. SpringerVerlag 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 smallgroup 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 smallgroup tutorials sessions.
Professionalism: Students work autonomously by developing and sustain effective approaches to learning, including time management, organisation, flexibility, creativity, collaboratively and intellectual integrity.
Study time
Type  Required  Optional 

Lectures  20 sessions of 1 hour (20%)  2 sessions of 1 hour 
Seminars  4 sessions of 1 hour (4%)  
Private study  66 hours (66%)  
Assessment  10 hours (10%)  
Total  100 hours 
Private study description
Weekly revision of lecture notes and materials, wider reading and practice exercises working on problem sets and preparing for the examination.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Problem set 1  2%  2 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Problem set 2  3%  2 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Problem set 3  2%  2 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Problem set 4  3%  2 hours 
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a wellprepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. 

Mathematical Methods for Statistics and Probability examination  90%  2 hours 
You will be required to answer all questions on this examination paper.

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  
You will be required to answer all questions on this examination paper.

Feedback on assessment
Individual feedback will be provided on problem sheets by class tutors.
Solutions and cohort level feedback will be provided for the examination
Students are actively encouraged to make use of office hours to build up their understanding, and to view all their interactions with lecturers and class tutors as feedback.
Courses
This module is Core for:
 Year 2 of USTAG302 Undergraduate Data Science
 Year 2 of USTAG304 Undergraduate Data Science (MSci)

USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 2 of G30A Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream)
 Year 2 of G30B Master of Maths, Op.Res, Stats & Economics (Econometrics and Mathematical Economics Stream)
 Year 2 of G30C Master of Maths, Op.Res, Stats & Economics (Operational Research and Statistics Stream)
 Year 2 of G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
 Year 2 of G300 Mathematics, Operational Research, Statistics and Economics
 Year 2 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 2 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 2 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics