By <Bayes theorem>, the <frailty distribution among survivors> has density
$$
g(u,t)
=\frac{\mathbb P(T>t\mid U=u)g(u)}{\overline S(t)}
=\{1+H_0(t)\}e^{-u\{1+H_0(t)\}},
\qquad u\geq0.
$$
It is an <exponential distribution> with rate $1+H_0(t)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 6 / b / Solution

In a <proportional frailty model>, <Bayes theorem> tilts the original density $g(u)$ among subjects surviving to $t$ to
$$
g(u\mid T>t)=\frac{e^{-uH_0(t)}g(u)}{\int_0^\infty e^{-vH_0(t)}g(v)\,dv}.
$$
This survival selection progressively favors smaller frailties.

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