 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-1/3i/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

The Shannon-Fano lengths are

(ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}) = (4, 3, 2, 2) .

One corresponding prefix-free assignment is

1 \mapsto 1010, 2 \mapsto 100, 3 \mapsto 00, 4 \mapsto 01 .

Its expected word length is

\overline{ℓ}_{SF} = \frac{1}{10} 4 + \frac{2}{10} 3 + \frac{3}{10} 2 + \frac{4}{10} 2 = 2.4 .

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