probability
A first course in probability. While I love probability, I found this course a little boring — though that's mostly my own doing: it's an area I'd spent a lot of time reading about on my own before Oxford, so I'd already covered most of the theory. The applications to random walks & branching processes, as well as the tougher questions on the problem sheets, were very fun.
The course was taught well, and I'll definitely take the Part A Probability course next year, which I'll hopefully enjoy more since the content will be new to me.
syllabus
Sample space, events, probability measure. Permutations and combinations, sampling with or without replacement. Conditional probability, partitions of the sample space, law of total probability, Bayes' Theorem. Independence.
Discrete random variables, probability mass functions, examples: Bernoulli, binomial, Poisson, geometric. Expectation, expectation of a function of a discrete random variable, variance. Joint distributions of several discrete random variables. Marginal and conditional distributions. Independence. Conditional expectation, law of total probability for expectations. Expectations of functions of more than one discrete random variable, covariance, variance of a sum of dependent discrete random variables.
Solution of first and second order linear difference equations. Random walks (finite state space only).
Probability generating functions, use in calculating expectations. Examples including random sums and branching processes.
Continuous random variables, cumulative distribution functions, probability density functions, examples: uniform, exponential, gamma, normal. Expectation, expectation of a function of a continuous random variable, variance. Distribution of a function of a single continuous random variable. Joint probability density functions of several continuous random variables (rectangular regions only). Marginal distributions. Independence. Expectations of functions of jointly continuous random variables, covariance, variance of a sum of dependent jointly continuous random variables.
Random sample, sums of independent random variables. Markov's inequality, Chebyshev's inequality, Weak Law of Large Numbers.