A8: Probability (2026-27)
Main content blocks
- 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 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). Reversibility, detailed balance equations. Random walks (including symmetric and asymmetric random walks on \(Z\), and symmetric random walks on \(Z^d\).
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). Reversibility, detailed balance equations. Random walks (including symmetric and asymmetric random walks on \(Z\), and symmetric random walks on \(Z^d\).
Section outline
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A couple of things to note about past exam papers:
- there was a slight change to the syllabus, starting with the 2024 paper. 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.
Here are some recent exam papers. (Both 2024 and 2026 include a question on reversibility.) Further revision materials may be added later in the year.
(a): 2026 exam paper. Solutions available.
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Problem sheet 1 and problem sheet 2 are now available. Sheets 3 and 4 will appear soon. Last year's sheets can be found on the 2025-2026 course page.