STAT4528: Semester 1, 2025

Overview

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory (based on measure theory) that was developed by Andrey Kolmogorov. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, conditional expectation, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and to thoroughly study discrete-time martingales. Originally used to model betting strategies, martingales are a powerful generalisation of random walks that allow us to prove fundamental results such as the Strong Law of Large Numbers or analyse problems such as the gambler's ruin. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability, stochastic systems or financial mathematics.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	None
Corequisites ?	None
Prohibitions ?	STAT4028
Assumed knowledge ?	STAT2X11 or equivalent and STAT3X21 or equivalent; that is, a good foundational knowledge of probability and some acquaintance with stochastic processes
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Michael Stewart, michael.stewart@sydney.edu.au
Lecturer(s)	Michael Stewart, michael.stewart@sydney.edu.au

Assessment

The census date for this unit availability is 31 March 2025

Details
Criteria

Type	Description	Weight	Due	Length
Supervised exam ?	Final exam Written examination	60%	Formal exam period	2 hours
Outcomes assessed:
Small continuous assessment	Weekly homework Weekly homework	40%	Multiple weeks	weekly
Outcomes assessed:
= AI allowed ?

Assessment summary

Weekly Homework: These are short exercises requiring students to digest lecture/tutorial material and provide written solutions. These will be set in most weeks of the semester. Artificial Intelligence (AI) tools may be used, however
- any such use must be acknowledged;
- AI output should be reviewed and verified;
- work which is clearly AI output but is either not acknowledged and/or not reviewed or verified may be heavily penalised.
Final exam: The exam will cover all material in the unit from both lectures and tutorials. The exam will have questions to answer. It is a 2h exam during the formal exam period.

Detailed information for each assessment would be given during the tutorial or can be found on Canvas

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name	Mark range	Description
High distinction	85 - 100	Representing complete or close to complete mastery of the material.
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery.
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
Fail	0 - 49	When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see guide to grades.

Use of generative artificial intelligence (AI) and automated writing tools

Except for supervised exams or in-semester tests, you may use generative AI and automated writing tools in assessments unless expressly prohibited by your unit coordinator.

For exams and in-semester tests, the use of AI and automated writing tools is not allowed unless expressly permitted in the assessment instructions.

The icons in the assessment table above indicate whether AI is allowed – whether full AI, or only some AI (the latter is referred to as “AI restricted”). If no icon is shown, AI use is not permitted at all for the task. Refer to Canvas for full instructions on assessment tasks for this unit.

Your final submission must be your own, original work. You must acknowledge any use of automated writing tools or generative AI, and any material generated that you include in your final submission must be properly referenced. You may be required to submit generative AI inputs and outputs that you used during your assessment process, or drafts of your original work. Inappropriate use of generative AI is considered a breach of the Academic Integrity Policy and penalties may apply.

The Current Students website provides information on artificial intelligence in assessments. For help on how to correctly acknowledge the use of AI, please refer to the  AI in Education Canvas site.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Support for students

The Support for Students Policy 2023 reflects the University’s commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy 2023. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

Weekly schedule

WK	Topic	Learning activity
Week 01	Introduction to Measure Theory	Lecture and tutorial (4 hr)
Week 02	Sigma Algebras and measurable functions	Lecture and tutorial (4 hr)
Week 03	Lebesgue Integrals	Lecture and tutorial (4 hr)
Week 04	Convergence	Lecture and tutorial (4 hr)
Week 05	Product Measures	Lecture and tutorial (4 hr)
Week 06	Central Limit Theorem	Lecture and tutorial (4 hr)
Week 07	Conditional Expectation	Lecture and tutorial (4 hr)
Week 08	Discrete Time Martingales	Lecture and tutorial (4 hr)
Week 09	Discrete Stopping Times	Lecture and tutorial (4 hr)
Week 10	Continuous Time Martingales	Lecture and tutorial (4 hr)
Week 11	Continuous Stopping Times and Local Martingales	Lecture and tutorial (4 hr)
Week 12	Brownian motion; semimartingales	Lecture and tutorial (4 hr)
Week 13	Revision week	Lecture and tutorial (4 hr)

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. Demonstrate a coherent and advanced knowledge of the concepts of measure theory and Lebesgue integration and how they provide a unified approach to a wide variety of problems arising in probability.
LO2. Communicate mathematical analyses and solutions to mathematical and practical problems in probability and related fields clearly in a variety of media to diverse audiences.
LO3. Apply characteristic function techniques to prove foundational theoretical results in probability.
LO4. Compare and contrast different forms of stochastic convergence.
LO5. Develop a theoretical tool set using martingales and Brownian motion for constructing solutions to a broad suite of problems in statistics, mathematical finance and other applied fields.
LO6. Devise solutions to novel mathematical problems in probability theory.
LO7. Understand the concept of martingale and its basic properties, and be able to recognise important examples of martingales arising Statistics, Finance and Probability theory
LO8. Be able to use the Optional Stopping Theorem in order to compute some important probabilities and expectations and understand its limitations

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

We now offer more smaller assignments to improve feedback to students.

Additional information

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

This unit of study outline was last modified on 12 Feb 2025.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI) and automated writing tools

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect

Current students

STAT4528: Probability and Martingale Theory

Semester 1, 2025 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI) and automated writing tools

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect