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ST318 Probability Theory

Throughout the 2020-21 academic year, we will be adapting the way we teach and assess your modules in line with government guidance on social distancing and other protective measures in response to Coronavirus. Teaching will vary between online and on-campus delivery through the year, and you should read the additional information linked on the right hand side of this page for details of how this will work for this module. The contact hours shown in the module information below are superseded by the additional information.You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.

All dates for assessments for Statistics modules, including coursework and examinations, can be found in the Statistics Assessment Handbook at http://go.warwick.ac.uk/STassessmenthandbook

ST318-15 Probability Theory

Academic year
20/21
Department
Statistics
Level
Undergraduate Level 3
Module leader
Paul Chleboun
Credit value
15
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry
Introductory description

This module runs in Term 2 and is available for students on a course where it is a listed option and as an Unusual Option to students who have completed one of the prerequisite modules.

Pre-requisite(s): ST342 Mathematics of Random Events OR MA359 Measure Theory.
Leads to: ST401 Stochastic Methods in Finance, ST402 Risk Theory, ST403 Brownian Motion, ST411 Dynamic Stochastic Control

Module web page

Module aims

This course aims to give the student a rigorous presentation of some fundamental results in measure theoretic probability and an introduction to the theory of discrete time martingales. In so doing it aims to provide a firm basis for advanced work on probability and its applications.

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.

  • Independence and zero-one laws
  • Modes of convergence for sequences of random variables
  • Limit theorems: law of large numbers (LLN) and central limit theorems (CLT)
  • Conditioning and discrete time martingales
Learning outcomes

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

  • understand the different modes of convergence for sequences of random and the relationship between these different modes;
  • state and prove the Central Limit Theorem via the method of characteristic functions and understand how this result can be applied;
  • understand some basic results on discrete -time martingales, including the martingale convergence theorem and optional stopping theorem, and show how these results can be used to obtain various characteristics of simple random walks.
  • understand the ideas relating to independence and zero-one laws and be able to apply these ideas in simple contexts;
Indicative reading list

View reading list on Talis Aspire

Subject specific skills

TBC

Transferable skills

TBC

Study time

Type Required Optional
Lectures 30 sessions of 1 hour (20%) 2 sessions of 1 hour
Tutorials 5 sessions of 1 hour (3%)
Private study 115 hours (77%)
Total 150 hours
Private study description

Weekly revision of lecture notes and materials, wider reading, practice exercises and preparing for examination.

Costs

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

Students can register for this module without taking any assessment.

Assessment group B1
Weighting Study time
2 hour examination (Summer) 100%

The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.

~Platforms - Moodle

Assessment group R
Weighting Study time
2 hour examination (September) 100%

The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.

~Platforms - Moodle

Feedback on assessment

Solutions and cohort level feedback will be provided for the examination.

Past exam papers for ST318

Courses

This module is Core for:

  • USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
    • Year 3 of G30A Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream)
    • Year 3 of G30D Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
  • USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
    • Year 3 of G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int
    • Year 3 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)
    • Year 4 of G30E Master of Maths, Op.Res, Stats & Economics (Actuarial and Financial Mathematics Stream) Int
    • Year 4 of G30H Master of Maths, Op.Res, Stats & Economics (Statistics with Mathematics Stream)

This module is Optional for:

  • Year 3 of UCSA-G4G1 Undergraduate Discrete Mathematics
  • Year 3 of UCSA-G4G3 Undergraduate Discrete Mathematics
  • Year 4 of UCSA-G4G2 Undergraduate Discrete Mathematics with Intercalated Year
  • USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
    • Year 3 of G300 Mathematics, Operational Research, Statistics and Economics
    • Year 4 of G300 Mathematics, Operational Research, Statistics and Economics

This module is Core option list A for:

  • Year 3 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
  • Year 4 of USTA-G1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)

This module is Option list A for:

  • UMAA-G105 Undergraduate Master of Mathematics (with Intercalated Year)
    • Year 3 of G105 Mathematics (MMath) with Intercalated Year
    • Year 5 of G105 Mathematics (MMath) with Intercalated Year
  • Year 3 of UMAA-G100 Undergraduate Mathematics (BSc)
  • UMAA-G103 Undergraduate Mathematics (MMath)
    • Year 3 of G103 Mathematics (MMath)
    • Year 4 of G103 Mathematics (MMath)
  • UMAA-G106 Undergraduate Mathematics (MMath) with Study in Europe
    • Year 3 of G106 Mathematics (MMath) with Study in Europe
    • Year 4 of G106 Mathematics (MMath) with Study in Europe
  • Year 4 of UPXA-GF14 Undergraduate Mathematics and Physics (with Intercalated Year)
  • Year 4 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
  • Year 3 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
  • Year 4 of USTA-GG17 Undergraduate Mathematics and Statistics (with Intercalated Year)
  • Year 4 of UMAA-G101 Undergraduate Mathematics with Intercalated Year
  • Year 3 of USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics
  • Year 4 of USTA-Y603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)

This module is Option list B for:

  • Year 1 of TMAA-G1PE Master of Advanced Study in Mathematical Sciences
  • Year 3 of USTA-G302 Undergraduate Data Science
  • Year 3 of USTA-G304 Undergraduate Data Science (MSci)
  • Year 4 of USTA-G303 Undergraduate Data Science (with Intercalated Year)

This module is Option list C for:

  • Year 3 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
  • USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
    • Year 3 of G30F Master of Maths, Op.Res, Stats & Economics (Econometrics and Mathematical Economics Stream) Int
    • Year 4 of G30F Master of Maths, Op.Res, Stats & Economics (Econometrics and Mathematical Economics Stream) Int

This module is Option list D for:

  • Year 4 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
  • Year 5 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated