 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-34/2/e/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

1/ (x lo g 10)

. A leading digit

i

corresponds to

i \leq X < i + 1

, so

P (D = i) = \frac{1}{lo g 10} \int_{i}^{i + 1} \frac{d x}{x} = lo g_{10} (1 + 1/ i), i = 1, \dots, 9.

These are exactly the Benford law probabilities. Continuous densities make endpoint conventions immaterial.

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