 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/27k/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The lifetime

L

is uniform on

[0, 3]

with mean

3/2

. The equilibrium age and excess densities are both

\frac{P ( L > x )}{E L} = \frac{2}{3} (1 - \frac{x}{3}) .

Hence, for

0 \leq x \leq 3

P (A_{t} \leq x), P (E_{t} \leq x) ⟶ \frac{2 x}{3} - \frac{x ^{2}}{9} .

The observed total lifetime is length-biased, so

P (A_{t} + E_{t} \leq x) ⟶ \frac{E [ L 1 _{L \leq x} ]}{E L} = \frac{x ^{2}}{9} .

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