Survival preferentially selects smaller frailties, so $\mathbb E[U\mid T\geq t]$ generally decreases with $t$. For the two-point distribution,
$$
\gamma(u,t)=
\frac{e^{-u\theta t}
[\delta(u-1/2)+\delta(u-3/2)]/2}
{[e^{-\theta t/2}+e^{-3\theta t/2}]/2}.
$$
At $t=0$ this equals $g(u)$, and
$$
\mathbb E[U\mid T\geq t]
=\frac{\tfrac12e^{-\theta t/2}+\tfrac32e^{-3\theta t/2}}
{e^{-\theta t/2}+e^{-3\theta t/2}}.
$$
It decreases from $1$ to $1/2$, demonstrating survivor selection toward the less frail class.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-207/6/b/solution