- Student Records
Programme & Unit Catalogues


MA40239: Discrete probability

Follow this link for further information on academic years Academic Year: 2019/0
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
Further information on unit levels Level: Masters UG & PG (FHEQ level 7)
Further information on teaching periods Period:
Semester 2
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this module you must take MA30125 or you must have taken MA20225 and have consulted the unit lecturer.
Further information on descriptions Description: Aims:
To introduce some discrete probabilistic structures with relevance to networks, statistical physics and communications - in particular, random graphs and percolation. To present some of the key results on these structures and the mathematical ideas and techniques behind these results. To acquaint the student with notions of phase transition, spatial and combinatorial methods in probability theory.

Learning Outcomes:
On completing the course, students should be able to:
* Describe the giant component phenomenon in random graphs and the phase transition phenomenon in percolation theory;
* Compute percolation thresholds on graphs such as trees;
* Perform simple computations on discrete probability structures (for example, mean subgraph counts on random graphs, or lattice animal counts);
* Apply tools such as Harris' inequality and branching process approximation;
* Appreciate ideas behind such theorems as the infinite cluster's uniqueness.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials)

Content:
Random Graphs: definition and motivation; small subgraphs; giant component and phase transition; maximum degree; clique number. Other aspects of random graphs such as: connectivity, chromatic number, bipartite matchings, sharp thresholds.
Percolation Theory: non-triviality of the phase transition; lattice animals, percolation on trees, uniqueness of the infinite component, properties and interpretation of the percolation probability. Other aspects of percolation theory such as: the critical value on the square lattice, self-avoiding walk, random resistor networks.
Further topics in discrete probability may be considered such as: invariant distributions, bounds and cut-off phenomena for Markov chain mixing times and examples thereof; entropy, noiseless coding, discrete memoryless channel in information theory.
Further information on programme availabilityProgramme availability:

MA40239 is Optional on the following programmes:

Department of Economics
  • UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
  • UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
  • UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
  • UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
  • RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
  • TSMA-AFM17 : MRes Statistical Applied Mathematics
  • TSMA-AFM08 : MSc Modern Applications of Mathematics
  • TSMA-AFM16 : MSc Statistical Applied Mathematics
  • USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
  • USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
  • USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
  • USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
  • USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
  • USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
  • USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
  • USMA-AFB05 : BSc(Hons) Statistics (Year 3)
  • USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
  • USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
  • USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
  • USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)

Notes: