Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 4 / b

Table of contents
- Solution b

Solution

For any coupling

π

of

X \sim P

and

Y \sim Q

and every

A \subseteq Ω

,

P (A) - Q (A) = P_{π} (X \in A, Y \in / A) - P_{π} (X \in / A, Y \in A) .

Its absolute value is at most

P_{π} (X \neq = Y)

. Taking the supremum over

A

and then the infimum over couplings proves

d_{TV} (P, Q) \leq in f_{π \in Π (P, Q)} P_{π} (X \neq = Y) .

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