# STATS 710 : Probability Theory

## Science

### Course Prescription

Fundamental ideas in probability theory; sigma-fields, laws of large numbers, characteristic functions, the Central Limit Theorem.

### Course Overview

This course will provide an introduction to probability theory, including the classical limit theorems of probability & statistics. It will cover fundamental theory, models and methods in probability whilst providing a solid mathematical foundation for research or advanced work in probability, statistical theory, statistical physics, or stochastic modelling in business, finance, economics, mathematical biology, computer science and engineering. Students taking this course may also take related courses such as STATS 720, 723, and 730 and MATHS 730, 764, and 769.

### Course Requirements

Prerequisite: STATS 310, 320 or 325

### 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. Apply the main results and techniques of probability theory. (Capability 1, 3 and 4)
2. Solve a variety of problems in probability using rigorous mathematical arguments. (Capability 1, 2, 3, 4 and 5)
3. Perform calculations and work effectively with sigma-algebras, probability measures, conditional expectations as random variables, and generating functions. (Capability 1, 3 and 4)
4. Apply the classical limit theorems, convergence theorems and main inequalities of probability. (Capability 1, 2, 3 and 4)
5. Distinguish between and determine the various types of convergence for random variables. (Capability 1, 2, 3 and 4)
6. Develop rigorous arguments and clear explanations, as appropriate. (Capability 1, 2, 3, 4 and 5)

### Assessments

Assessment Type Percentage Classification
Assignments 40% Individual Coursework
Final Exam 60% Individual Examination
1 2 3 4 5 6
Assignments
Final Exam

### Key Topics

Topics include: Probability measures, event spaces, sigma-algebras, Borel sets, continuity of probability. The Fundamental model of uniform probability measure on [0,1] and Lebesgue-Borel Theorem. Independence. Borel-Cantelli Lemmas. Random variables. Expectation. Monotone and Dominated Convergence Theorems. Inequalities for expectations. Conditional expectation as a random variable. Discrete time martingales. Generating functions, including Characteristic functions and Laplace transforms. Convergence of generating functions. Types of convergence. Sequences of independent random variables. Weak and Strong Laws of Large Numbers, and the Central Limit Theorem. Other topics, models or applications of probability.

### Learning Resources

We suggest "Knowing the Odds: An Introduction to Probability" by John Walsh as a recommended, but not required, textbook.

### Special Requirements

N/A

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures, a 1 hour tutorial, 2 hours of reading and thinking about the content and 4 hours of work on assignments and/or test preparation.

### Digital 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.

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.

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.

### 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 at http://disability.auckland.ac.nz

### 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: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### 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.

Based on student feedback in previous years, this year's course will introduce weekly tutorials, in which students can practice skills from the course with supervision and support from the instructors.

### 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 (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### 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 you may be asked to submit your 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. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 13/07/2020 11:47 a.m.