 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-60/2/i/a/solution

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

Every conditional output distribution is a permutation of

q

. Since Shannon entropy is invariant under a permutation, its value is the same for every input symbol. Thus, for any input distribution,

H (Y ∣ X) = x \sum p (x) H (Y ∣ X = x) = H (q) = - j = 1 \sum 3 q_{j} lo g_{2} q_{j} .

All Shannon entropies in this solution are in bits. A different fixed logarithm base changes every entropy and capacity by the same constant factor.

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