ST22910 Probability for Mathematical Statistics
Introductory description
This module introduces core concepts in Probability and Statistics that are needed for further modules in both Probability and Statistics.
Prerequisites: ST118 Probability 1, ST119 Probability 2, and ST117 Introduction to Statistical Modelling, or equivalents.
This module is core for students with their home department in Statistics. It is not available to other students, for whom ST232/ST233 (Introduction to Mathematical Statistics) is provided as an alternative.
This module is a prerequisite for ST230 Mathematical Statistics and a number of ST3 and ST4 modules.
Module aims
The firstyear modules “Probability 1 and 2” and “Introduction to Statistical Modelling” introduce, respectively, a solid foundation in the mathematical formalisation of probability, and the fundamentals of statistical modelling and statistical inference. This module provides a bridge between these foundational topics, and the more advanced modules in year 3 and 4. Most of these advanced topics require the use of probability distributions over more than one random variable. This module focuses on the mathematical properties of such distributions, and the consequences of these properties in the context of statistical modelling and inference.
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 introduces the systematic study of the theory of mathematical statistics.
 Statistical motivation for studying sample averages and joint distributions.
 Definitions of joint, marginal and conditional distributions (discrete and continuous cases). Independent random variables. Conditional expectation and properties. Conditional variance and properties.
 Multivariate Normal/Gaussian distribution, its properties and distributions related to it. Simulating Gaussian distributions.
 Properties of finite sums and averages of random variables, expectation and variance of sample means, focusing on the Normal/ Gaussian distribution case.
 Weak law of large numbers and the central limit theorem.
Learning outcomes
By the end of the module, students should be able to:
 Derive properties of finite and infinite sums and averages of random variables, including the central limit theorem. Describe its significance in statistics.
 Demonstrate knowledge of joint and conditional probability distributions, conditional expectation and conditional variance.
 State/derive and use the properties of multivariate Normal/Gaussian distribution and distributions related to Normal/Gaussian distribution.
 Communicate solutions to problems accurately with structured and coherent arguments.
Indicative reading list
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
Subject specific skills
Demonstrate knowledge of key mathematical and statistical concepts, both explicitly and by applying them to the solution of mathematical problems.
Select and apply appropriate mathematical and/or statistical techniques
Create structured and coherent arguments communicating them in written form.
Transferable skills
Problemsolving 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 staff office hours.
Professionalism: Students work autonomously by developing and sustaining effective approaches to learning, including timemanagement, organisation, flexibility, creativity, collaboration and intellectual integrity.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (20%) 
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. 

Probability for Mathematical Statistics 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.
Cohortlevel feedback will be available on the exam.
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