STAT3021: Semester 1, 2022

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Units STAT3021 Semester 1 2022 [Normal day]

Unit of study_

STAT3021: Stochastic Processes

Overview

A stochastic process is a mathematical model of time-dependent random phenomena and is employed in numerous fields of application, including economics, finance, insurance, physics, biology, chemistry and computer science. This unit will establish basic properties of discrete-time Markov chains including random walks and branching processes. This unit will derive key results of Poisson processes and simple continuous-time Markov chains. This unit will investigate simple queuing theory. This unit will also introduce basic concepts of Brownian motion and martingales. Throughout the unit, various illustrative examples are provided in modelling and analysing problems of practical interest. By completing this unit, you will develop an essential basis for further studies stochastic analysis, stochastic differential equations, stochastic control, financial mathematics and statistical inference.

Details

Academic unit	Mathematics and Statistics Academic Operations
Unit code	STAT3021
Unit name	Stochastic Processes
Session, year ?	Semester 1, 2022
Attendance mode	Normal day
Location	Camperdown/Darlington, Sydney
Credit points	6

Enrolment rules

Prohibitions ?	STAT3911 or STAT3011 or STAT3921 or STAT4021
Prerequisites ?	STAT2X11
Corequisites ?	None
Assumed knowledge ?	Students are expected to have a thorough knowledge of basic probability and integral calculus
Available to study abroad and exchange students	Yes

Teaching staff and contact details

Coordinator	Qiying Wang, qiying.wang@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Take-home short release)	Exam written exam (online)	60%	Formal exam period	2 hours
Outcomes assessed:
Assignment	Assignment 1 written paper work	5%	Week 04	days
Outcomes assessed:
In-semester test (Record+)	Quiz consist of short answer questions.	25%	Week 07 Due date: 04 Apr 2022 at 12:00	50 minutes
Outcomes assessed:
Assignment	Assignment 2 written paper work	10%	Week 12	days
Outcomes assessed:
= Type B in-semester exam ? = Type D final exam ?

Detailed information for each assessment can be found on Canvas

Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

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 sydney.edu.au/students/guide-to-grades.

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.

Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

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.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

Weekly schedule

WK	Topic	Learning activity
Week 01	Preliminaries: moment generating function, joint distribution, conditional probability and conditional expectation, random sum and basic concepts of stochastic processes.	Lecture and tutorial (4 hr)
Week 02	Markov chain: transition probabilities, Chapman-Kolmogorov equations, and classification of states.	Lecture and tutorial (4 hr)
Week 03	Markov Chain: Periodicity, recurrence and transience, positive and null recurrences.	Lecture and tutorial (4 hr)
Week 04	Markov Chain: limit distribution, stationary distribution, absorption probability and mean return time.	Lecture and tutorial (4 hr)
Week 05	Random walk and Branching processes: Gambler's ruin problem, expected duration, extinction probability	Lecture and tutorial (4 hr)
Week 06	Random walk and Branching processes: expected duration, extinction probability, Weeks 1-6 revision	Lecture and tutorial (4 hr)
Week 07	Quiz 1, basic properties of Poisson distribution and exponential distribution	Lecture and tutorial (4 hr)
Week 08	Poisson processes: definition, interarrival and arrival (waiting) times, conditional distribution of arrival time.	Lecture and tutorial (4 hr)
Week 09	Poisson processes: splitting and merging a Poisson process, nonhomogeneous Poisson process and compound Poisson process	Lecture and tutorial (4 hr)
Week 10	Continuous-time Markov chains: definition and basic properties, the Embedded Markov Chain and the generator matrix, forward and backward Equations, stationary and limit distributions.	Lecture and tutorial (4 hr)
Week 11	Simple queuing theory: the $M/M/1$ queue, the $M/M/1$ queue with finite capacity and the $M/M/k, k\ge 1,$ queue	Lecture and tutorial (4 hr)
Week 12	Introduction to BM and martingale and Weeks 7-12 revision	Lecture and tutorial (4 hr)
Week 13	Revision, Assignments revisited and pass exams.	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. explain and be able to apply the fundamentals of probability theory and stochastic processes
LO2. construct a discrete-time Markov chain and identify its transition probability matrix from practical problem settings
LO3. explain and be able to apply limit theorems of discrete-time Markov chains and use those to identify and interpret their stationary distribution
LO4. explain Gambler's ruin problem and calculate extinction probability
LO5. explain the basic properties of the Poisson process and use these to solve problems
LO6. construct a Poisson process and identify its parameter from practical problem settings
LO7. construct a continuous-time Markov chain and identify its generator from practical problem settings
LO8. explain the length in the queue and solve simple waiting time problems
LO9. explain definitions of Brownian and martingales
LO10. write proofs and apply the theory in diverse applications.

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

Closing the loop

No changes have been made since this unit was last offered

Additional information

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Follow safety instructions in your manual and posted in laboratories
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First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

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Current students

Overview

Details

Enrolment rules

Teaching staff and contact details

Assessment

Assessment criteria

Late submission

Special consideration

Academic integrity

Weekly schedule

Study commitment

Learning outcomes

Graduate qualities

Outcome map

Closing the loop

Additional information

Work, health and safety

Disclaimer

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