# STATS 225 : Probability: Theory and Applications

## Science

### 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).

### Course Requirements

Prerequisite: B+ or higher in ENGGEN 150 or ENGSCI 111 or STATS 125, or a B+ or higher in MATHS 120 and 130 Corequisite: 15 points from ENGSCI 211, MATHS 208, 250

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Understand and apply the main concepts underlying probability theory (Capability 1, 2 and 4)
2. Work effectively with probabilities, expectations, and random variables (Capability 1, 2 and 3)
3. Apply generating function methods to solve various problems in probability (Capability 1, 2 and 3)
4. Organise a mathematical calculation for coherence and readability (Capability 1, 2, 3, 4 and 5)
5. Produce clearly reasoned proofs in probability at an appropriate level of difficulty (Capability 1, 2, 3, 4 and 5)
6. Understand and use fundamental stochastic models and their basic properties to solve problems in probability (Capability 1, 2, 3, 4 and 5)
7. Solve a variety of problems in probability (Capability 1, 2, 3 and 5)

### Assessments

Assessment Type Percentage Classification
Coursework - Assignments 30% Individual Coursework
Final Exam 60% Individual Examination
Coursework - Test 10% Individual Coursework
1 2 3 4 5 6 7
Coursework - Assignments
Final Exam
Coursework - Test

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

### Tuākana

Tuākana Science 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
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

### 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

### Special Requirements

Attending tutorials to improve problem solving skills is highly recommended.

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
• 3-5 hours of work on assignments and/or test preparation

### Delivery Mode

#### Campus Experience

Participation is strongly recommended at all scheduled activities (including tutorials) to successfully complete this course.

Lectures will be available as recordings. Other learning activities such as in-person tutorials will not typically be available as recordings, but the course may additionally include live online tutorials. 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 on campus is not required for any mid-semester test, which will be online where applicable. The final exam will normally be on campus.

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

### Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes 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.

• "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.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Some potential future improvements in the light of student feedback received:
• Trying to improve tutorial engagement or other ways of getting peer interaction, especially online.
• Balance the weekly exercise sheet lengths/difficulties better eg. one or two were a bit longer. This might involve cutting some questions that were more general/fun rather than on essential material (or mark as ‘extras for fun only’), and provide more indications of essential/optional/harder questions on each sheet.
• Ongoing improvements to content to adjust difficulty and quantity. Consider whether to simplify or cut some topics (eg. other types of generating functions, central limit theorem proof)

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 against online source material using computerised detection mechanisms.

### Class Representatives

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.

### 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

### Special Circumstances

If your ability to complete assessed coursework 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.

### 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.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

### 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 .

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course 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.

Published on 01/11/2022 09:37 a.m.