 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/3i/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Let

q = P (X = B)

. The merged channel is a Binary Z-channel: its second output occurs with probability

q /2

, while the output has one bit of uncertainty only when

X = B

h_{2}

I (X; Y) = h_{2} (q /2) - q .

At an interior maximum,

0 = \frac{d I}{d q} = \frac{1}{2} lo g_{2} \frac{1 - q /2}{q /2} - 1,

(2 - q) / q = 4

and

q = 2/5

. Consequently

C = h_{2} (1/5) - \frac{2}{5} = lo g_{2} \frac{5}{4} bits .

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