 Solution

The population survivor function is the probability that a randomly selected person, averaging over their unobserved frailty, has not experienced the event by time $t$. Here the conditional hazard $U\theta$ is constant, so the conditional survival is $e^{-U\theta t}$. For $\theta >0$, the uniform frailty density equals one on its interval, giving the uniform frailty survival mixtureAt $t=0$ its continuous extension is $\overline{S}(0)=1$, since the numerator is $\theta t+O(t^2)$. This is the Laplace transform of the uniform frailty density at argument $\theta t$. It is not $e^{-\theta t}$: averaging individual survivor functions does not equal exponentiating the mean hazard. If $\theta =0$, every individual hazard is zero and $\overline{S}(t)=1$.