Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 7 / a

The joint probability mass function is $p_{X,Y}(x,y)=\mathbb{P}(X=x,Y=y)$ for $(x,y)\in S^2$. It is nonnegative and its sum over $S^2$ equals one. Summing over one coordinate gives the other variable's marginal distribution.

For $\mathbb{P}(X=x)>0$, the conditional probability mass function isThe positive denominator makes the expression well-defined, and its sum over $y$ is one.

A partition of a sample space is a finite or countable family of pairwise disjoint events $B_i$ whose union is the whole sample space. For an integrable random variable, the partition form of the law of total expectation isZero-probability cells contribute zero and are omitted because elementary event-conditioning on them is undefined. The formula also holds with extended nonnegative values when $X\ge 0$. Integrability makes the signed series legitimate in the finite-expectation case.