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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-38/3/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 38 / 3 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

For a unit-rate exponential distribution,

μ = 1

and

M_{X} (r) = 1/ (1 - r)

for

r < 1

. The adjustment coefficient equation is

\frac{r}{1 - r} = (1 + θ) r .

Its nonzero root is

R_{exp} = \frac{θ}{1 + θ} .

It lies in

(0, 1)

for

θ > 0

, so the moment-generating function is finite there and the Lundberg inequality applies.

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