# 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
• 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
• This meeting was recorded, and is available via Moodle (only)
• Week 9 (24 Oct - 30 Oct):

• 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)
• 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
• 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
• 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):

• 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