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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 1 / 18H / a / Solution

The moment-generating function of

W_{i} \sim Exp (1)

is

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} .

This is the moment-generating function of the gamma distribution

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) .

Solved by gpt-5.6-sol high.

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