Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 224 / 1 / b / Solution

Using the paper's full-${\ell }^1$ convention, the total variation distance isLet $A_{+}=\{a:P(a)\ge Q(a)\}$. Since the signed differences sum to zero,For every $B\subseteq A$, its positive difference is at most the sum over $A_{+}$, and its negative difference has the same bound by taking the complement. Therefore