A8: Probability (2021-22)
Main content blocks
- Lecturer: Profile: Matthias Winkel
Moment generating functions and applications. Statements of the continuity and uniqueness theorems for moment generating functions. Characteristic functions (definition only). Convergence in distribution and convergence in probability. Weak law of large numbers and central limit theorem for independent identically distributed random variables. Strong law of large numbers (proof not examinable).
Discrete-time Markov chains: definition, transition matrix, n-step transition probabilities, communicating classes, absorption, irreducibility, periodicity, calculation of hitting probabilities and mean hitting times. Recurrence and transience. Invariant distributions, mean return time, positive recurrence, convergence to equilibrium (proof not examinable), ergodic theorem (proof not examinable). Random walks (including symmetric and asymmetric random walks on \(Z\), and symmetric random walks on \(Z^d\).
Poisson processes in one dimension: exponential spacing’s, Poisson counts, thinning and superposition.
Section outline
-
The Lecture Notes 📒 have 7 chapters.There are 45 Lecture Videos of varying length, probably best accessed through the Panopto panel to the right of this page (or at the bottom if your browser window is narrow - if this does not work at first, try the link to the Panopto folder behind "Lecture Videos" here or in the General panel above).
Apart from the video on Chapter 1, which reviews Prelims Probability, each video covers either a section or a subsection. They are all named accordingly for ease of reference. You will find further navigation within the videos under Contents.
What I've written during the videos is available as a Video Script 📜. There are only two videos, where I also use slides, First Slides 🖼️ in the first video on Chapter 1 and Last Slides 🖼️ in the last video on Section 7.5.
There are four problem sheets. 📄 Sheet 1 covers Chapters 1-2, 📄 Sheet 2 covers Chapters 3-4, 📄 Sheet 3 covers Chapter 5 and 📄 Sheet 4 covers Chapters 6-7.
[There is also a full set of live Lecture Recordings 📽️ from Michaelmas Term 2019, but this link may not give the right login screen if you're not already appropriately logged in, so try the Old Repository 📦.]
-
These appendices contain a discussion of the theoretical background to the Prelims Probability course, establishing and exploiting connections to the Analysis courses in Prelims and Part A. This additional material is non-examinable in Part A Probability.
-
This appendix includes a discussion of some more theoretical background to the Part A Probability course. This additional material is non-examinable in Part A Probability.