### MAS3322 : Applied Probability

• Offered for Year: 2016/17
• Module Leader(s): Dr Philip Ansell
• Owning School: Mathematics & Statistics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 1 Credit Value: 10 ECTS Credits: 5.0
Code Title
MAS2304

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#### Aims

To develop skills in probabilistic reasoning and to gain familiarity with some of the main techniques involved in the analysis of random systems. To develop skills in the analysis of a variety of simple queues.

Module Summary

How can we predict the chance of a gambler winning given a certain strategy? On average how much will he win? Uncertainty is a central feature of almost every real-life problem, and questions of this nature arise naturally in many applications ranging from economics and finance through to engineering, sports betting and queues. In the first part of this course, we shall discover how certain probabilistic techniques can be used to model and analyse systems or phenomena that evolve randomly over time.
In the second part of the course, we will look in detail at the analysis of queues. Queues occur in many situations. For example in telecommunications, traffic engineering, computing and in the design of factories, shops and hospitals all of which are subject to congestion. We will discuss how we construct and analyse models for systems of queues. Crucial questions are how congested any such system is likely to become over time and how we design such systems optimally.

#### Outline Of Syllabus

Review of probability ideas: conditioning arguments. Random walks: solution of gambler's ruin problems, returns to the origin. Markov chains: definition and examples, Chapman-Kolmogorov equations, classification of states, notions of transience and recurrence, stationary distributions. Components of a queue; Poisson process; the M/M/1 queue; Birth-Death models; the M/M/s system; optimum number of servers; embedded Markov process; the M/G/1 queue; simulation models; Renewal processes.

#### Learning Outcomes

##### Intended Knowledge Outcomes

Students will gain knowledge of issues arising in the study of the simple random walk, the gambler's ruin problem. Students will become familiar with the notion of Markov chains,and will gain some understanding of their probabilistic description. Students will have a working knowledge of the Poisson process. Students will gain an understanding of renewal processes.

##### Intended Skill Outcomes

Students will be able to classify states of Markov chains and find the stationary distributions(s). Students will be able to produce steady-state analyses of a variety of simple queues using balance equations and will be able to use embedded Markov chains to analyse M/G/1 queues

• Cognitive/Intellectual Skills
• Numeracy : Assessed
• Self Management
• Personal Enterprise
• Problem Solving : Assessed
• Interaction
• Communication
• Written Other : Assessed

#### Teaching Methods

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion28:0016:00Assignments
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision lectures
Guided Independent StudyAssessment preparation and completion111:0011:00Revision for unseen Exam
Guided Independent StudyAssessment preparation and completion11:301:30Unseen Exam
Scheduled Learning And Teaching ActivitiesLecture61:006:00Problem class
Scheduled Learning And Teaching ActivitiesLecture221:0022:00Formal lectures
Guided Independent StudyAssessment preparation and completion24:008:00Revision for class test
Guided Independent StudyAssessment preparation and completion21:002:00Class test
Scheduled Learning And Teaching ActivitiesPractical21:002:00Practical class
Scheduled Learning And Teaching ActivitiesDrop-in/surgery61:006:00Drop-ins in the lecture room
Guided Independent StudyIndependent study119:3019:30Studying, practising and gaining understanding of course material
Guided Independent StudyIndependent study41:004:00Assignment review
Total100:00
##### Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the studentsâ€™ abilities at applying the theory to solving problems. Drop-ins are used to identify and resolve specific queries raised by students and to allow students to receive individual feedback on marked work. Practicals are used both for solution of problems and work requiring extensive computation and to give insight into the ideas/methods studied. Office hours provide an opportunity for more direct contact between individual students and the lecturer.

#### Assessment Methods

The format of resits will be determined by the Board of Examiners

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination901A80Unseen examination
##### Exam Pairings
Module Code Module Title Semester Comment
MAS8322Applied Probability1N/A
##### Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Assignments
Prob solv exercises1M10Coursework tests
##### Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. Written assignments (approximately 2 pieces of work of approximately equal weight) and two coursework tests (in class) allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; this is thus formative as well as summative assessment.

#### General Notes

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Disclaimer: The information contained within the Module Catalogue relates to the 2016/17 academic year. In accordance with University Terms and Conditions, the University makes all reasonable efforts to deliver the modules as described. Modules may be amended on an annual basis to take account of changing staff expertise, developments in the discipline, the requirements of external bodies and partners, and student feedback. Module information for the 2017/18 entry will be published here in early-April 2017. Queries about information in the Module Catalogue should in the first instance be addressed to your School Office.