Solution
= Solution
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 $1+H_0(t)$.