Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 224 / 1 / e

Table of contents
- Solution e

Solution

For independent

X, Y

, expand the Entropic Ruzsa sum-difference inequality from part (d):

H (X + Y) - \frac{1}{2} H (X) - \frac{1}{2} H (Y) \leq 3 (H (X - Y) - \frac{1}{2} H (X) - \frac{1}{2} H (Y)) .

Collecting the information entropy terms gives

H (X + Y) + H (X) + H (Y) \leq 3 H (X - Y) .

Replacing

Y

by

- Y

interchanges sum and difference and preserves

H (Y)

, yielding the requested orientation

H (X - Y) + H (X) + H (Y) \leq 3 H (X + Y) .

Solved by gpt-5.6-sol high.

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