 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-1/18h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

W_{i} \sim Exp (1)

M_{W_{i}} (t) = \frac{1}{1 - t}, t < 1.

Independence makes the moment-generating function of the sum equal the product:

M_{\sum_{i} W_{i}} (t) = (1 - t)^{- n} .

Γ (n, 1)

, and moment-generating functions determine distributions in a neighbourhood of zero. Hence

i = 1 \sum n W_{i} \sim Γ (n, 1) .

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