Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 7 / a

For each integer $x$, the events $\{X=x,Y=y\}$, indexed by the integers $y$, are disjoint and their union is $\{X=x\}$. Therefore countable additivity gives the marginal probability mass functionThe nonnegative terms make this sum well-defined, even when the support is infinite.

The random variables are independent random variables if the events determined by one are independent of the events determined by the other. For integer-valued variables, this is equivalent to the factorizationIndeed, independence gives the factorization by taking singleton events. Conversely, summing it over $x\in A$ and $y\in B$, for any subsets $A,B\subseteq \mathbb{Z}$, gives $\mathbb{P}(X\in A,Y\in B)=\mathbb{P}(X\in A)\mathbb{P}(Y\in B)$. Thus the singleton condition implies independence of all events in question.

Interpret existence of the expectations as finite expected values, equivalently $\mathbb{E}|X|<\infty$ and $\mathbb{E}|Y|<\infty$. First justify the summations rather than assuming that a signed double series can be rearranged. Independence and Tonelli theorem giveConsequently $XY$ is an integrable random variable, and Fubini theorem permits summing the signed terms in either order:If extended infinite expectations are allowed instead, the phrase “exist” needs additional care: expressions such as $0\cdot \infty$ do not define an expectation. The finite-expectation convention avoids that ambiguity.