APP MTH 3001 - Applied Probability III
North Terrace Campus - Semester 1 - 2022
General Course Information
Course Code APP MTH 3001 Course Applied Probability III Coordinating Unit School of Mathematical Sciences Term Semester 1 Level Undergraduate Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Prerequisites (MATHS 1012 and MATHS 2103) or (MATHS 2201 and MATHS 2202) or (MATHS 2106 and MATHS 2107) Assumed Knowledge Knowledge of Markov Chains such as would be obtained from MATHS 2103 Course Description 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 tool kit 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, homogeneous 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.
Course Coordinator: Jasper Barr
The full timetable of all activities for this course can be accessed from Course Planner.
Course Learning Outcomes
- demonstrate understanding of the mathematical basis of discrete-time Markov chains and martingales
- demonstrate the ability to formulate discrete-time Markov chain models for relevant practical systems
- demonstrate the ability to apply the theory developed in the course to problems of an appropriate level of difficulty
- demonstrate the ability to conduct a group project applying the theory developed in this course
- develop an appreciation of the role of applied probability in mathematical modelling
- demonstrate skills in communicating mathematics orally and in writing
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s)
Attribute 1: Deep discipline knowledge and intellectual breadth
Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.
Attribute 2: Creative and critical thinking, and problem solving
Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.
Attribute 3: Teamwork and communication skills
Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.
Attribute 4: Professionalism and leadership readiness
Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.
Attribute 5: Intercultural and ethical competency
Graduates are responsible and effective global citizens whose personal values and practices are consistent with their roles as responsible members of society.
Attribute 8: Self-awareness and emotional intelligence
Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.
Recommended ResourcesThere are many good books on probability and statistics in the Barr Smith Library, with the following texts being recommended for this course.
1. "Probability and Random Processes" (Oxford, 2001).
2. "Introduction to Probability Models" by Sheldon Ross (Academic Press, 2010).
3. "An introduction to Stochastic Modelling" by Taylor and Karlin (Academic Press, 1998).
4. "A First Course in Stochastic Processes" by Karlin and Taylor (Academic Press, 1975).
5. "Elementary Probability Theory with Stochastic Processes" by Kai Lai Chung (Springer-Verlag, 1975).
6. "An Introduction to Probability Theory and its Applications" by Feller (Wiley, 1968).
7. "Introduction to Stochastic Models" by Roe Goodman (2nd edition, Dover, 2006).
8. "Markov chains" by James Norris (Cambridge, 1997).
For other texts on probability and statistics, try browsing books with call numbers beginning with 519.2.
Online LearningThis course uses MyUni exclusively for providing electronic resources, such as notes, videos, quizzes, assignments and solutions et cetera.
Learning & Teaching Activities
Learning & Teaching ModesEach week, lecture notes will be provided, designed to be read in advance of viewing videos. Videos will consist of the lecturer explaining key material and examples from the lecture notes.
The videos will be supported by two weekly classes, a tutorial and a workshop. In the workshop the lecturer will guide you through the week’s material, incorporating active learning exercises, whilst the tutorial is focused on practicing problems to reinforce this learning.
Five written assignments, five online quizzes and a mid-semester test provide the assessment opportunities for students to strengthen their understanding of the theory and their skills in applying it, and gauge and demonstrate their progress and understanding.
The group project allows students to develop their teamwork and communication skills, and apply their knowledge to a challenging problem in a practical environment.
Interaction with the lecturer is encouraged during contact hours.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Weekly online materials 12 weeks 68 Tutorials 12 24 Workshops 12 12 Mid-semester test 1 1 Online quizzes 5 5 Assignments 5 20 Group project 1 25 Total 155
Learning Activities Summary
Topics Schedule 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.
Tutorials will cover the content of the previous week, while the first tutorial in Week 1 will focus on revision.
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Assessment task When due Weighting Learning outcomes Examination Examination period 35% All Assignments Weeks 2, 4, 6, 10 and 12 20% All Group project Week 10 20% All Mid-semester test Week 8 15% All Quizzes Weeks 3, 5, 7, 9 and 11 10% All
Assessment Related RequirementsAn aggregate score of 50% is required in order to pass this course.
Assessment task Set Due Weighting Assignment 1 Week 1 Week 2 4% Quiz 1 Week 3 Week 3 2% Assignment 2 Week 3 Week 4 4% Quiz 2 Week 5 Week 5 2% Assignment 3 Week 5 Week 6 4% Quiz 3 Week 7 Week 7 2% Mid-semester test Week 8 Week 8 15% Quiz 4 Week 9 Week 9 2% Assignment 4 Week 9 Week 10 4% Group project Week 2 Week 10 20% Quiz 5 Week 11 Week 11 2% Assignment 5 Week 11 Week 12 4% Exam Exam period 35%
SubmissionAssignments must be submitted on time and online via MyUni. Late assignments will not be accepted. 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 and online via MyUni. 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.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description 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.
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