Live/In-person session diary 2022

General information

We may attempt to live stream, the live in-person sessions. Please note that it is UCL’s expectation that students attend in-person. The Zoom link is available here. Not all sessions will be recorded or streamed.

By UCL week

• Week 6 (3 Oct - 9 Oct):

  • Introductions, first meeting.
  • Introduction to R, Python
  • Uniform random variables and generating random variables
  • (Weak) law of large numbers
  • This meeting was recorded, and is available via Moodle (only)

• Week 7 (10 Oct - 16 Oct):

  • We discussed rejection sampling and introduced the concept of coupling.
  • Random triangles worksheet
    • solutions

• Week 8 (17 Oct - 23 Oct):

  • We discussed the homework; in particular, random walks on graphs and reversibility.
  • Playing with a two-by-two Markov chain and symbolic computing in Python
    • worksheet
      • solutions
    • This meeting was recorded, and is available via Moodle (only)

• Week 9 (24 Oct - 30 Oct):

  • We discussed how Doeblin’s coulping was the basis of our convergence theorems for Markov chains, and we also introduced Poisson point processes on finite volumes.
  • Playing with data from a Markov chain
    • worksheet
      • solutions
  • Shiny app for Poisson on a disc
    • Source
  • Poisson and WW2
  • Animal versus human poll, what Americans think
  • This meeting was recorded, and is available via Moodle (only)

• Week 10 (31 Oct - 6 Nov):

  • We discussed the expectations for ICA1, and the module in general. We also went over a homework exercise on the uniform distribution, and introduced continuous Markov chains, using holding times.
  • Reinforced random walk (non-Markovian processes)
    • solutions
    • optional materials to help understand trapping
    • optional materials to help understand simple random walk

• Week 11 (7 Nov - 13 Nov): Reading break

• Week 12 (14 Nov - 20 Nov):

  • We discussed the inspection paradox in the context of Poisson processes
  • We discussed the renewal theory and the strong Markov property in the context of Markov chains
  • worksheet
    • solutions
  • This meeting was recorded, and is available via Moodle (only)

• Week 13 (21 Nov - 27 Nov):

  • We discussed the excessive time distribution and its connection to stationary increments.
  • We discussed how the Blackwell renewal theorem hold of Poisson processes, and how it is easier for renewal processes with stationary increments:
    • We argued that in the stationary case, we have \(m(t+h) - m(t) = \mathbb{E}N(h) = m(h)\); I tried to make an argument that \(\mathbb{E}N(h) = h/\mu\) just using the fact that \(\lim_{h \to \infty}m(t)/t = 1/\mu\), which is not correct; a more involved argument is given in the notes.
  • worksheet
    • solutions
  • My imagination of what the stock market is like
  • This meeting was recorded, and is available via Moodle (only)

• Week 14 (28 Nov - 4 Dec):

  • We introduced Little’s theorem
  • worksheet
    • solutions
  • Euclid on incommensurables and incomparables
  • This meeting was recorded, and is available via Moodle (only)

• Week 15 (5 Dec - 11 Dec):

  • We discussed \(M/M/1\) queues and Burke’s theorem
  • We discussed ICA2
  • We reviewed an old ICA3 from 2021
  • This meeting was recorded, and is available via Moodle (only)

• week 16 (12 Dec - 16 Dec):

  • We discussed solutions to ICA 1.
  • We discussed Borel-Cantelli and the strong law of large numbers link
  • This meeting was recorded, and is available via Moodle (only)

Other notes

• First term (Monday 26 September 2022 – Friday 16 December 2022)

• week 12: (14 Nov - 20 Nov; 2022): ICA1 due, 18 November

• week 24: (6 Feb - 12 Feb; 2023): ICA3 due, 9 February

• week 27: (27 Feb - 5 Mar; 2023): ICA2 due, 3 March

Version: 16 December 2022