= Solution
A <frailty model> uses an unobserved nonnegative random <hazard multiplier> to represent variation in individual risk. In a <proportional frailty model>, conditional on $U=u$ the <cumulative hazard function> is $uH_0(t)$, giving survivor $e^{-uH_0(t)}$. Averaging over the <frailty random variable> by the <law of total probability> yields
$$
\boxed{S(t)=\int_0^\infty e^{-uH_0(t)}g(u)\,du
=\widetilde g(H_0(t)),}
$$
the <Laplace transform> of the frailty distribution evaluated at the baseline integrated hazard. This also applies to atoms by integration against the probability measure; the printed Dirac expression is not an ordinary continuous density.
In the given two-atom case, the <Dirac delta function> notation assigns probabilities $1/4$ and $3/4$ to $U=4/7$ and $U=8/7$. These have mean one, fixing the frailty scale relative to the <baseline hazard>. With $H_0(t)=7t/4$, the two conditional integrated hazards are $t$ and $2t$. Therefore
$$
\boxed{S(t)=\tfrac14e^{-t}+\tfrac34e^{-2t}=w(t).}
$$
The <frailty distribution among survivors> shifts toward the smaller multiplier $4/7$ over time, reproducing the declining population hazard and limit one without changing either individual's constant conditional hazard.
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