BSc Computer Science and Mathematics with Industrial Experience

Year of entry: 2024

Course unit details:
Foundations of Modern Probability

Course unit fact file
Unit code MATH37021
Credit rating 10
Unit level Level 3
Teaching period(s) Semester 1
Available as a free choice unit? No

Overview

The Law of Large Numbers (LLN) and the Central Limit Theorem (CLT) are two most important/ fundamental results in classical and modern probability theory with numerous applications.  

Extending the intuitive understanding and partial proofs developed in Probability 1 & 2, the unit builds on methods from mathematical analysis to cultivate a rigorous understanding and derive complete proofs of fundamental results in probability theory, focusing in particular on the LLN and the CLT. 

Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Modern Probability MATH20722 Anti-requisite Compulsory
Mathematical Foundations & Analysis MATH11121 Pre-Requisite Compulsory
Probability and Statistics 2 MATH27720 Pre-Requisite Compulsory
"Cannot take MATH37021 if taken 20722; pre-reqs MATH11121 MATH27720"

Aims

The unit aims to:

- Introduce students to fundamental concepts, methods, and tools in modern probability  

 theory in a systematic and rigorous way (by presenting proofs), offering robust theoretical  

 foundations for subsequent studies in probability theory and related areas (such as Financial  

 Mathematics).

- Guide students to develop a rigorous understanding and derive complete proofs of  

 fundamental results in probability theory, such as the Law of Large Numbers and the  

 Central Limit Theorem, while extending the intuitive understanding and partial proofs  

 of these results developed in Probability 1 & 2. 

Learning outcomes

On the successful completion of the course, students will be able to:  


  • Calculate probabilities and expected values using more advanced probabilistic methods such as the second Borel-Cantelli lemma or Kolmogorov's 0-1 law.
  •  
    State and use fundamental inequalities (Markov, Jensen, Holder, Minkowski) and modes of convergence (almost sure, in probability, in distribution, in mean). 

  • State and use monotone convergence theorem, dominated convergence theorem, and Fatou’s lemma. 

  • State and derive the law of large numbers and the central limit theorem in a variety of theoretical and applied settings. 

Teaching and learning methods

2 contact hours per week, divided between review class and examples class. Most of content delivered via online videos (approximately 75 minutes per week).

Assessment methods

Method Weight
Written exam 80%
Written assignment (inc essay) 20%

Feedback methods

Preparing questions on video content 
Feedback provided during Review class

Weekly exercise sheet (submit attempted solutions) 
Feedback provided during Examples class

Midterm test 
General feedback provided after test is marked.


Final exam 
General feedback provided after exam is marked. 

Recommended reading


[1] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press.

[2] Shiryaev, A. N. (1996). Probability. Springer-Verlag. 

Study hours

Scheduled activity hours
Lectures 11
Practical classes & workshops 11
Independent study hours
Independent study 78

Teaching staff

Staff member Role
Xiong Jin Unit coordinator

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