 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-215/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

(X, Y)

P (X \in A) = μ (A)

and

P (Y \in A) = ν (A)

. No independence is required.

For every set

A

, the difference of the two indicator functions vanishes when

X = Y

. Consequently

∣ μ (A) - ν (A) ∣ = E (1_{A} (X) - 1_{A} (Y)) \leq P (X \neq = Y) .

Maximizing over

A

∥ μ - ν ∥_{TV} \leq P (X \neq = Y) .

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