Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 215 / 1 / b

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- Solution b

Solution

A coupling of probability distributions is a joint probability distribution of random variables

(X, Y)

with the specified marginal distributions:

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

gives the coupling inequality for total variation:

∥ μ - ν ∥_{TV} \leq P (X \neq = Y) .

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