# Faculty of Mathematics

Please take this page in conjunction with the Part III Guide to Courses Probability and Finance section.

For many Part III Probability courses, you will need a significant amount of Measure Theory. If you are not yet familiar with it, some solid preparation work over the summer will be necessary for you to keep up in lectures.

• Basic Probability
• Measure Theory: for Advanced Probability, Stochastic Calculus and Applications (on which other courses rely, see prerequisites listed in Guide to Courses);

## Basic Probability

The relevant Cambridge undergraduate course is Ia Probability.

Reality check

• Notions of independence, covariance, and correlation between random variables.
• Conditional probability and Bayes' theorem.
• The axiomatic formulation of probability theory (i.e. probability spaces, sigma-algebras, the notion of random variables as functions on a probability space).
• Markov's inequality and Chebyshev's inequality.
• Convergence in distribution, in probability and almost-surely.
• Asymptotic results such as the strong/weak law of large numbers and the central limit theorem.
• Moment generating functions and characteristic functions.

You should also have some level of familiarity with commonly-encountered probability distributions, such as the multivariate normal distribution, the Poisson distribution, and the binomial distribution. You can check your level of these topics by going through the Example Sheets from the Probability course, to be found under the link above. We recommend that you look only at the main questions, excluding any "extra", "additional" or "starred" questions.

Tools from Analysis are often used in Part III Probability courses, and you should be familiar with concepts from standard undergraduate Analysis, such as convergence and continuity. The relevant Cambridge undergraduate courses are Ia Analysis I and Ib Analysis II. You can check your level of these topics by going through the Example sheets from these two courses. We recommend that you look only at the main questions, excluding any "extra", "additional" or "starred" questions.

### Useful books and resources

• Prof. Weber's lecture notes from 2015.
• More very good resources on Prof Weber's page.

## Measure Theory

This section is on the Analysis and PDEs page.