Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 1 / b

Solution

Let

V, V^{'}, U

be mutually independent, with

V, V^{'}

H (V + V^{'} + U) + H (U) \leq H (V + U) + H (V^{'} + U) .

Independent addition cannot decrease discrete entropy, because

H (V + V^{'} + U) \geq H (V + V^{'} + U ∣ U) = H (V + V^{'})

. Hence

H (V + V^{'}) + H (U) \leq H (U + V) + H (U + V^{'}) .

Take

U

to have the distribution of

- X

and take

V, V^{'}

to be independent copies of

X

, all mutually independent. Then

U + V

and

U + V^{'}

both have the distribution of

X - Y

, whereas

V + V^{'}

has the distribution of

X + Y

and

H (U) = H (X)

. Hence

H (X + Y) + H (X) \leq 2 H (X - Y),

H (X + Y) - H (X) \leq 2 {H (X - Y) - H (X)} .

The denominator is nonnegative because conditioning on

Y

recovers

X

from

X - Y

, so

H (X - Y) \geq H (X - Y ∣ Y) = H (X)

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