|Semester 1 Credit Value:||10|
|MAS2304||Foundations of Probability|
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.
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.
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.
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.
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
|Graduate Skills Framework Applicable:||Yes|
|Guided Independent Study||Assessment preparation and completion||1||1:30||1:30||Unseen Exam|
|Scheduled Learning And Teaching Activities||Lecture||6||1:00||6:00||Problem class|
|Guided Independent Study||Assessment preparation and completion||2||4:00||8:00||Revision for class test|
|Scheduled Learning And Teaching Activities||Lecture||22||1:00||22:00||Formal lectures|
|Guided Independent Study||Assessment preparation and completion||2||1:00||2:00||Class test|
|Guided Independent Study||Assessment preparation and completion||2||8:00||16:00||Assignments|
|Scheduled Learning And Teaching Activities||Lecture||2||1:00||2:00||Revision lectures|
|Guided Independent Study||Assessment preparation and completion||1||11:00||11:00||Revision for unseen Exam|
|Scheduled Learning And Teaching Activities||Practical||2||1:00||2:00||Practical class|
|Scheduled Learning And Teaching Activities||Drop-in/surgery||24||0:00||0:00||Office Hours in a staff office|
|Scheduled Learning And Teaching Activities||Drop-in/surgery||6||1:00||6:00||Drop-ins in the lecture room|
|Guided Independent Study||Independent study||1||19:30||19:30||Studying, practising and gaining understanding of course material|
|Guided Independent Study||Independent study||4||1:00||4:00||Assignment review|
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 students’ 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.
The format of resits will be determined by the Board of Examiners
|Written Examination||90||1||A||80||Unseen examination|
|Module Code||Module Title||Semester||Comment|
|Prob solv exercises||1||M||10||Assignments|
|Prob solv exercises||1||M||10||Coursework tests|
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.
Note: The Module Catalogue now reflects module information relating to academic year 14/15. Please contact your School Office if you require module information for a previous academic year.
Disclaimer: The University will use all reasonable endeavours to deliver modules in accordance with the descriptions set out in this catalogue. Every effort has been made to ensure the accuracy of the information, however, the University reserves the right to introduce changes to the information given including the addition, withdrawal or restructuring of modules if it considers such action to be necessary.