Course: A8: Probability (2023-24) | Mathematical Institute

Lecturer: Profile: James Martin

Course information

Course Term: Michaelmas

Course Lecture Information: 16 lectures

Course Overview:

The first half of the course takes further the probability theory that was developed in the first year. The aim is to build up a range of techniques that will be useful in dealing with mathematical models involving uncertainty. The second half of the course is concerned with Markov chains in discrete time and Poisson processes in one dimension, both with developing the relevant theory and giving examples of applications.

Course Synopsis:

Continuous random variables. Jointly continuous random variables, independence, conditioning, functions of one or more random variables, change of variables. Examples including some with later applications in statistics.

Moment generating functions and applications. Statements of the continuity and uniqueness theorems for moment generating functions. Characteristic functions (definition only). Convergence in distribution, convergence in probability, and almost sure convergence. 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). Reversibility, detailed balance equations. Random walks (including symmetric and asymmetric random walks on \(Z\), and symmetric random walks on \(Z^d\).

Section outline

Select section General

General

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- Select activity Announcements
  
  Announcements Forum
- Select activity Revision materials
  
  Revision materials Assignment
  
  A couple of things to note about past exam papers:
  - the syllabus has changed slightly this year. Poisson processes are no longer covered, and there is new material on reversibility.
  - the papers from 2020 and 2021 had a different style since they were taken under open-book conditions.
  I've prepared four practice exam papers, involving closed-book-suitable questions from previous years that don't involve Poisson processes, and adding a couple of new questions touching on reversibility.
  Solutions for two of the papers are available for students to download.
  (a): 2017 exam paper. This paper is suitable in its entirety.
  (b): Practice paper 1. Q2 and Q3 from 2015, and one new question.
  (c): Practice paper 2. Q1 from 2019, Question M5 from paper AO2 of 2009, and one new question. Solutions available.
  (d): Practice paper 3. Q1 (edited) and Q3 from 2018, Q2 from 2020. Solutions available.
  Of course many other questions from previous years are also suitable! Just avoid any material on Poisson processes if you want to stick to this year's syllabus.
Select section Course Materials

Course Materials
- Select activity All four problem sheets
  
  All four problem sheets Assignment
- Select activity Complete lecture notes (version of 25 November 2023)
  
  Complete lecture notes (version of 25 November 2023) File

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Section outline

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Course Materials