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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 27K / b / Solution

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} .

Solved by gpt-5.6-sol high.

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