Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 4 / a

The joint probability mass function isIt is nonnegative and its sum over $S\times S$ is one. Summing over one coordinate gives the other marginal probability mass function.

For $\mathbb{P}(X=x)>0$, the conditional probability mass function isThe denominator is the sum of the numerator over all possible $y$, so these conditional probabilities sum to one.

A partition of a sample space is a finite or countable family $(A_j)$ of pairwise disjoint events whose union is the whole sample space. If $X$ is integrable, the law of total expectation statesThis follows by summing $X{1}_{A_j}$ over the partition of a sample space; absolute integrability justifies the countable sum. For nonnegative $X$, the identity also holds with infinite expectations. Zero-probability parts contribute zero without requiring a conditional value on them.