Bachelor of Science (BSc)
BSc Actuarial Science and Mathematics
- Typical A-level offer: A*AA including specific subjects
- Typical contextual A-level offer: A*AB including specific subjects
- Refugee/care-experienced offer: A*BB including specific subjects
- Typical International Baccalaureate offer: 37 points overall with 7,6,6 at HL, including specific requirements
Fees and funding
Fees
Tuition fees for home students commencing their studies in September 2025 will be £9,535 per annum (subject to Parliamentary approval). Tuition fees for international students will be £34,500 per annum. For general information please see the undergraduate finance pages.
Policy on additional costs
All students should normally be able to complete their programme of study without incurring additional study costs over and above the tuition fee for that programme. Any unavoidable additional compulsory costs totalling more than 1% of the annual home undergraduate fee per annum, regardless of whether the programme in question is undergraduate or postgraduate taught, will be made clear to you at the point of application. Further information can be found in the University's Policy on additional costs incurred by students on undergraduate and postgraduate taught programmes (PDF document, 91KB).
Scholarships/sponsorships
The University of Manchester is committed to attracting and supporting the very best students. We have a focus on nurturing talent and ability and we want to make sure that you have the opportunity to study here, regardless of your financial circumstances.
For information about scholarships and bursaries please visit our undergraduate student finance pages and our Department funding pages .
Course unit details:
Markov Processes
Unit code | MATH37011 |
---|---|
Credit rating | 10 |
Unit level | Level 3 |
Teaching period(s) | Semester 1 |
Available as a free choice unit? | No |
Overview
Markov chains are stochastic processes with the special property of "given the present, the future is independent of the past". Many real-life situations can be modelled by such processes and this course is concerned with their mathematical analysis. It by-passes the measure-theoretic considerations necessary for the development of a general theory of stochastic processes.
Pre/co-requisites
Unit title | Unit code | Requirement type | Description |
---|---|---|---|
Probability and Statistics 2 | MATH27720 | Pre-Requisite | Compulsory |
Aims
To develop the idea that processes evolving randomly in time can be modelled mathematically in terms of sequences or families of dependent random variables.
Learning outcomes
On completion of this course unit students should be able to:
- classify the states of a discrete time Markov chain;
- calculate the stationary and limiting distributions of discrete time Markov chains;
- state and prove results about recurrence, transience and periodicity of discrete time Markov chains;
- write down and solve the forward equations for simple birth- death processes;
- apply continuous time Markov chain theory to the modelling of queues;
- state and prove results about the limiting behaviour of continuous time Markov chains.
Future topics requiring this course unit
The material of this course may be helpful in understanding the 4th year courses on stochastic calculus and Brownian motion.
Syllabus
1.Discrete time Markov chains:
Review of necessary probability theory. [2]
Definition of Markov chain. Homogeneity. (1-step) transition probabilities. Transition diagrams. Examples including Ehrenfest diffusion model. The Chapman-Kolmogorov equations. Matrix form. Accessibility. Closed/irreducible sets. Periodicity. Stationary distributions. Positive recurrence, null recurrence, transience. Random walk examples. Convergence to stationary distribution. Discussion of different methods of proof (e.g. Markov's method for finite state space; Doeblin's coupling; renewal type argument). [8]
2.Continuous time Markov chains:
Theoretical treatment at level of Karlin and Taylor (see below). Revision of Poisson process. Pure birth/birth death processes. [8]
3.Applications:
Queues. M/M/1. Queue length and waiting time distribution. M/M/s. Variable arrival and service rates. Machine interference. [4]
Assessment methods
Method | Weight |
---|---|
Other | 20% |
Written exam | 80% |
- Coursework: two hours weighting 20%
- End of semester examination: weighting 80%
Feedback methods
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
Recommended reading
- D. R. Stirzaker, Stochastic Processes and Models, Oxford University Press, 2005.
- A. N. Shiryaev, Probability, Springer-Verlag, 1996.
- S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, Academic Press, 1975.
- D. R. Stirzaker, Elementary Probability, Cambridge University Press, 2003.
- G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford Univ. Press, 1992.
Study hours
Scheduled activity hours | |
---|---|
Lectures | 12 |
Tutorials | 12 |
Independent study hours | |
---|---|
Independent study | 76 |
Teaching staff
Staff member | Role |
---|---|
Jose Pedraza Ramirez | Unit coordinator |
Additional notes
The independent study hours will normally comprise the following. During each week of the taught part of the semester:
• You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently
• You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week
• There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises
• In some weeks you may be preparing coursework or revising for mid-semester tests
Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.
The remaining independent study time comprises revision for and taking the end-of-semester assessment.
The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.