= 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 mixture>
$$
\overline S(t)=\int_{1/2}^{3/2}e^{-u\theta t}\,du
=\boxed{\frac{e^{-\theta t/2}-e^{-3\theta t/2}}{\theta t}},\qquad t>0.
$$
At $t=0$ 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 $\overline S(t)=1$.
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