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- UCAS course code
- GG41
- UCAS institution code
- M20
BSc Computer Science and Mathematics with Industrial Experience
Year of entry: 2024
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Course unit details:
Foundations of Modern Probability
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 |
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 |