Consider two random variables $X$ and $Y$ which take values in a finite set $S$. Let $p_{X, Y}$ represent their joint probability mass function (p.m.f.) and let $p_{X}$ and $p_{Y}$, respectively, be the marginal p.m.f.'s of $X$ and $Y$, respectively. Which of the choices below is always equal to
\[\max _{A \subseteq S}|\operatorname{Pr}(X \in A)-\operatorname{Pr}(Y \in A)| ?\]
- $\frac{1}{2} \sum_{n \in S}\left|p_{X}(n)-p_{Y}(n)\right|$
- $\frac{1}{2} \sum_{n_{1}, n_{2} \in S: n_{1} \neq n_{2}} p_{X, Y}\left(n_{1}, n_{2}\right)$
- $\frac{1}{2} \sum_{n \in S} p_{X, Y}(n, n)$
- $\frac{1}{2} \sum_{n_{1}, n_{2} \in S: n_{1} \neq n_{2}}\left|p_{X}\left(n_{1}\right)-p_{Y}\left(n_{2}\right)\right|$
- None of the above