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Overview

Course Prescription

Covers the fundamentals of probability through theory, methods, and applications. Topics should include the classical limit theorems of probability and statistics known as the laws of large numbers and central limit theorem, conditional expectation as a random variable, the use of generating function techniques, and key properties of some fundamental stochastic models such as random walks, branching processes and Poisson point processes.

Course Overview

This course will provide an introduction to probability through some classical theories, useful techniques, and stochastic models fundamental to many applications. It should provide a solid mathematical foundation for more advanced courses in probability or mathematical statistics, and is important for anyone who wants to advance to honours or masters level in these areas. 

Before taking this course students should ideally have a basic background in probability (at least Grade B+ in one of STATS 125, ENGGEN 150, or ENGSCI 111) as well as good mathematics (at least Grade B+ in MATHS 120 and MATHS 130, or equivalent).  Some advanced mathematical topics should also be studied at the same time as, or before, taking this course (at least one of ENGSCI 211, MATHS 208, or MATHS 250).  

This course will also provide excellent preparation for more advanced courses in probability (such as STATS 325, STATS 710, STATS 720, or STATS 723).

Key Topics

Topics may include: 

  • Probability measures. Event spaces. Borel sigma-algebra. Properties of probabilities, including continuity of probability
  • Fundamental model of Uniform probability measure on [0,1] and Lebesgue-Borel Theorem (statement)
  • Independence
  • Borel-Cantelli Lemmas to determine finite or infinite number of successes in sequences of events. 
  • Random variables
  • Expectation. Monotone Convergence Theorem (statement)
  • Conditional probability and expectation. Conditional expectation with respect to a random variable
  • Generating functions techniques: PGFs, MGFs, Characteristic functions and Laplace transforms. Properties and convergence of generating functions
  • Types of convergence: Convergence almost surely, in probability, convergence in distribution, and convergence  Weak Law of Large Numbers. Strong Law of Large Numbers (proof of special case). Central Limit Theorem (sketch proof)
  • Random walks: First return times. First passage times. Gambler's ruin. Reflection principle. Ballot Theorem. Recurrence of random walks
  • Branching processes: discrete-time Galton-Watson process, extinction probabilities, population size
  • Poisson processes: characterisations, inter-arrival times, gamma distributions, thinning and conditional uniformity. Poisson point processes (PPPs) on R^n. Examples of PPPs with non-constant intensities

Workload Expectations

This course is a standard 15-point course and students are expected to spend a total of 150 hours per semester on each 15-point course that they are enrolled in. Students are expected to spend 10 hours per week working on this course during each of the 12 teaching weeks, plus an additional 30 hours overall in preparation for tests/final examinations (150 hours in total).

For this course, a typical weekly workload includes:

  • 3 hours of lectures
  • 1-hour tutorial
  • 1-3 hours of reading and thinking about the content
  • 3-5 hours of work on assignments and/or test preparation

Course Prerequisites, Corequisites and Restrictions

Prerequisite
Corequisite

Locations and Semesters Offered

LocationSemester
City

Teaching and Learning

Campus Experience

Participation is strongly recommended at all scheduled activities to successfully complete this course. Particularly, participation in tutorials is a vital part of your learning in this course. Students are expected to attend in person, on campus, to participate in this tutorial.

Lectures will be available as recordings. Other learning activities such as in-person tutorials will not typically be available as recordings. Online Q&A help will also be available via Piazza on the course Canvas site. All course materials will be made available online via Canvas.

Coursework may be submitted online via Canvas. Attendance in person will be required for any on campus test and exam.

The activities for the course will be scheduled as a standard weekly timetable over the 12-week teaching period of the semester.

Learning Resources

Taught courses use a learning and collaboration tool called Canvas to provide students with learning materials including reading lists and lecture recordings (where available). Please remember that the recording of any class on a personal device requires the permission of the instructor.

Additional Information on Learning Resources

Recommended Reading:

  • "Probability: an introduction" by G.R.Grimmett and D. Welsh (Oxford University Press). This is a good introductory book. Available to read online from UoA library
  •  "Probability and Random Processes" by Geoffrey Grimmett and David Stirzaker (Oxford University Press). This is a very useful book that covers many topics in probability, including some more advanced topics. Available in the UoA library.

Copyright

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.


You may copy the course content for the purposes of private study or research, but you may not upload onto any third-party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

Similarly, research students must meet the University’s expectations of good research practice. This requires:

  • Honesty - in all aspects of research work
  • Accountability - in the conduct of research
  • Professional courtesy and fairness – in working with others
  • Good stewardship – on behalf of others
  • Transparency – of research process and presentation of results
  • Clarity - communication to be understandable, explainable and accessible

For more information on the University’s expectations of academic integrity, please see the Academic Conduct section of the University policy hub.

Disclaimer

Elements of this outline may be subject to change. The latest information about taught courses is made available to enrolled students in Canvas.

Students may be asked to submit assessments digitally. The University reserves the right to conduct scheduled tests and examinations online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.


Assessment and Learning Outcomes

Additional Information on Assessment

A mark of at least 40% in the final exam (& 50% overall) is required to pass this course.

Special Requirements

Attending tutorials to improve problem solving skills is highly recommended.

Course Learning Outcomes

CLO #OutcomeProgramme Capability Link
1
2
3
4
5
6
7

Assessments

Assessment TypeAssessment PercentageAssessment Classification

Assessment to CLO Mapping

Assessment Type1234567

Student Feedback, Support and Charter

Student Feedback

Feedback on taught courses is gathered from students at the end of each semester through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback and respond with summaries and actions. Your feedback helps teachers to improve the course and its delivery for future students. In addition, class Representatives in each class can take feedback to the department and faculty staff-student consultative committees.

Additional Information on Student Feedback

Changes in the light of student feedback received:

  • Changes to the tutorialTrying to improve tutorial engagement or other ways of getting peer interaction, especially online.
  • Weekly exercise sheet lengths/difficulties have been rebalanced to give a more specific indication of the workload - some sheets have been split, and there is now a more explicit split of essential material versus optional/harder questions on each sheet.
  • Ongoing improvements to content to adjust difficulty and types of topics covered.

Class representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Tuākana

Tuākana is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at Tuakana Learning Communities.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website.

Wellbeing

We all go through tough times during the semester, or see our friends struggling. There is lots of help out there - please see the Support Services page for information on support services in the University and the wider community.

Special Circumstances

If your ability to complete assessed work is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due. If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page. This should be done as soon as possible and no later than seven days after the affected test or exam date.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter.

Student Academic Complaints and Disputes

Students with concerns about teaching including how a course is delivered, the resources provided, or supervision arrangements, have the right to express their concerns and seek resolution. The university encourages informal resolution where possible, as this is quicker and less stressful. For information on the informal and formal complaints processes, please refer to the Student Academic Complaints Statute in the Student Policies and Guidelines section of the Policy Hub.