Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 6 / a

For equal point masses at $1/2$ and $3/2$ with $h_0=\theta$,Its initial value is $\theta$ because $\mathbb{E}U=1$, while $\bar{h}(t)\to \theta /2$ as the more frail survivors are progressively depleted.

For $U\sim \mathrm{Exponential}(1)$,Again $\bar{h}(0)=\theta$ because $\mathbb{E}U=1$, but now $\bar{h}(t)\to 0$ as $t\to \infty$.