University of Sheffield, MAS275 Probability Modelling

Welcome to the web pages for MAS275 Probability Modelling. Course notes, exercises and solutions will appear here, in PDF format.

The lectures for this course are on Tuesdays at 12 in Hicks Lecture Theatre 7 and Fridays at 11 in Hicks Lecture Theatre 1.

Tutorials will be held every other week, starting in the second week of the course. They will be Tuesdays at 16:00; Wednesdays at 12:00; or Thursdays at 12:00 depending which group you are in; you can find out which group you are in and the name of your group's tutors at the SoMaS tutorial groups page. This will also tell you where the classes are.

Homework will be set to be handed in for weeks 3, 5, 7, 9 and 11: the folders will be left outside my office (I22 Hicks) from 11am on Tuesday and you should hand the work in by 2pm on Wednesday. The marking will be for feedback rather than for formal assessment.

To contact the lecturer, email t.heaton@sheffield.ac.uk. My office is I22, on the fifth floor of the Hicks Building. My office hour for the course is 12:00 - 13:00 on Fridays.

A discussion board for this module will be available on MOLE2.

Assessment for the course will be by a 2 hour closed book exam.

The notes are divided into six sections. Notes for each section are available from the links below:

- Introduction and Markov chains. Notes, slides.
- Renewal processes. Notes, slides.
- Long-run behaviour of Markov chains. Notes, slides.
- Google PageRank. Notes, slides
- Hitting times and probabilities. Notes, slides
- Poisson processes. Notes, slides

There is also some extra material which may be useful.

- Revision of material on series, including some revision exercises
- Summary of notation for section 2
- Proofs of some facts about null recurrent Markov chains

Slides for Markov Monopoly (Example 21)

Link to Page and Brin paper describing Google PageRank

Exercise booklet for the course. This covers the whole course.

- Chapter 1: Markov Chains
- Chapter 2: Renewal Processes
- Chapter 3: Limiting Behaviour of Markov Chains
- Chapter 4: Google
- Chapter 5: Hitting times and probabilities
- Chapter 6: Poisson Processes

- Sample R code for question 25 (simulating patterns in coin tossing)
- Sample R code for question 38(b) (simulating convergence of Markov chains)

Last updated 2 May 2019.