APP MTH 7065 - Applied Probability

North Terrace Campus - Semester 1 - 2014

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, originating in the fifteenth and sixteenth centuries. This course aims to provide a basic toolkit for modelling and analyzing 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.

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Course Details

Course Code	APP MTH 7065
Course	Applied Probability
Coordinating Unit	Applied Mathematics
Term	Semester 1
Level	Postgraduate Coursework
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Prerequisites	MATHS 1012
Assumed Knowledge	Markov Chains as taught in MATHS 2103
Assessment	ongoing assessment 30%, exam 70%

Course Staff

Course Coordinator: Professor Nigel Bean

Course Timetable

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

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
develop an appreciation of the role of applied probability in mathematical modelling
demonstrate skills in communicating mathematics orally and in writing

最新糖心Vlog Graduate Attribute	Course Learning Outcome(s)
Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised.	1,2,3,4
The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner.	2,3,5
An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems.	2,3,4,5
Skills of a high order in interpersonal understanding, teamwork and communication.	5
A proficiency in the appropriate use of contemporary technologies.	3
A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life.	1,2,3,4,5
An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities.	3,4,5

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.

Workload

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

**Lecture 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.

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.

The 最新糖心Vlog'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 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	20%	All
Group project	Formative and summative	Week 10	10%	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	4%
Assignment 2	Week 3	Week 4	4%
Assignment 3	Week 5	Week 6	4%
Assignment 4	Week 9	Week 10	4%
Assignment 5	Week 11	Week 12	4%
Group Project	Week 3	Week 10	10%

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. Late project reports will not be accepted. Project reports will be retained by the lecturer but will be assessed before the end of the teaching period prior to examinations and may be viewed by arrangement with the lecturer.

Course Grading

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 .

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Maintained by:	Course Outlines Administrator

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