## APP MTH 7065 - Applied Probability

### North Terrace Campus - Semester 1 - 2019

Many processes in the real world involve some random variation superimposed on a deterministic structure. For example, the experiment of flipping a coin is best studied by treating the outcome as a random one. Mathematical probability has its origins in games of chance with dice and cards, from the fifteenth and sixteenth centuries. This course aims to provide a basic toolkit for modelling and analysing discrete-time problems in which there is a significant probabilistic component. We will consider Markov chain examples in the course including population branching processes (with application to genetics), random walks (with application to games), and more general discrete time examples using Martingales. Topics covered are: basic probability and measure theory, discrete time Markov chains, hitting probabilities and hitting time theorems, population branching processes, inhomogeneous random walks on the line, solidarity properties and communicating classes, necessary and sufficient conditions for transience and positive recurrence, global balance, partial balance, reversibility, Martingales, stopping times and stopping theorems with a link to Brownian motion.

• General Course Information
##### Course Details
Course Code APP MTH 7065 Applied Probability Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week Y MATHS 2103 or (MATHS 2201 and MATHS 2202) Knowledge of Markov chains such as would be obtained from MATHS 2103 ongoing assessment, exam
##### Course Staff

Course Coordinator: Angus Lewis

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
1. demonstrate understanding of the mathematical basis of discrete-time Markov chains and martingales
2. demonstrate the ability to formulate discrete-time Markov chain models for relevant practical systems
3. demonstrate the ability to apply the theory developed in the course to problems of an appropriate level of difficulty
4. demonstrate the ability to conduct a group project applying the theory developed in this course
5. develop an appreciation of the role of applied probability in mathematical modelling
6. demonstrate skills in communicating mathematics orally and in writing

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
1,2,3,4
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
1,2,3,4
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
4,5,6
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
3,4,5,6
Intercultural and ethical competency
• adept at operating in other cultures
• comfortable with different nationalities and social contexts
• able to determine and contribute to desirable social outcomes
• demonstrated by study abroad or with an understanding of indigenous knowledges
4,6
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
4
• Learning Resources
None
##### Recommended Resources
There are many good books on probability and statistics in the Barr Smith Library, with the following texts being recommended for this course.

1. "Introduction to Probability Models" by Sheldon Ross (Academic Press, 2010).
2. "An introduction to Stochastic Modelling" by Taylor and Karlin (Academic Press, 1998).
3. "A First Course in Stochastic Processes" by Karlin and Taylor (Academic Press, 1975).
4. "Elementary Probability Theory with Stochastic Processes" by Kai Lai Chung (Springer-Verlag, 1975).
5. "An Introduction to Probability Theory and its Applications" by Feller (Wiley, 1968).
6. "Introduction to Stochastic Models" by Roe Goodman (2nd edition, Dover, 2006).

For other texts on probability and statistics, try browsing books with call numbers beginning with 519.2.
##### Online Learning
All assignments, tutorials, handouts and solutions, where appropriate, will be made available on MyUni as the course ensues.

Recordings of lectures will also be available on MyUni following each lecture, for those who are unable to attend due to other commitments and for revision purposes.

Please don't hesitate to email the lecturer should anything be missing.
• Learning & Teaching Activities
##### Learning & Teaching Modes
The lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures, but lectures will be recorded to help with occasional absences and for revision purposes. In fortnightly tutorials, students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Lectures 30 90 Tutorials 5 20 Assignments 5 25 Group project 1 20 Total 155
##### Learning Activities Summary
 Week 1 Basic probability theory Sample space and events. Laws of large numbers and the central limit theorem and their interpretation. Week 2 Basic probability theory. Algebras and sigma-algebras of events and probability measure. Week 3 Discrete time Markov chains Definition of a discrete time Markov chain (DTMC). Random walks. Week 4 Discrete time Markov chains Hitting probabilities and hitting times. Classification of states. Week 5 Discrete time Markov chains Recurrence and transience. Week 6 Discrete time Markov chains Irreducible DTMCs. Branching processes. Periodicity. Week 7 Discrete time Markov chains Limiting behaviour. Long term behaviour and global balance. Week 8 Discrete time Markov chains. Partial balance. Time reversal and reversibility. Week 9 Martingales Definition of a martingale. Fair games, branching processes and random walks. Week 10 Martingales Stopping times and optional stopping theorem. Dominated martingales and Optional stopping times. Week 11 Martingales Two dimensional random walks. Identifying martingales. Week 12 Martingales and Brownian motion Sub-martingales, super-martingales and construction of martingales. Motivation and definition of Brownian motion with examples. Review.

The first tutorial in Week 3 covers material from the previous two weeks and other material that should be considered revision. Tutorials in Weeks 5, 7, 9 and 11 cover the material of the previous few weeks.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
 Assessment task Task type When due Weighting Learning outcomes Examination Summative Examination period 70% All Assignments Formative and summative Weeks 2, 4, 6, 10 and 12 15% All Group project Formative and summative Week 10 15% All
##### Assessment Related Requirements
An aggregate score of 50% is required in order to pass this course.
##### Assessment Detail
 Assessment task Set Due Weighting Assignment 1 Week 1 Week 2 3% Assignment 2 Week 3 Week 4 3% Assignment 3 Week 5 Week 6 3% Assignment 4 Week 9 Week 10 3% Assignment 5 Week 11 Week 12 3% Group Project Week 2 Week 10 15%
##### Submission
Assignments must be submitted on time with a signed assessment cover sheet attached to the assignment. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. In such cases, documentation is required and the lecturer must be notified as soon as possible.

The final written project report must be submitted on time with an appropriate signed assessment cover sheet attached to the report itself. You must also submit a PDF version of the report and all source code via email to the lecturer. Late project reports will not be accepted. Project reports will be assessed before the end of the teaching period prior to examinations and will be returned to the group.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
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