= Solution
Use <Bayes theorem> with the individual <survivor function>. Writing $z=\theta t^2/2$, the <frailty distribution among survivors> has density
$$
g(u\mid T>t)=\frac{e^{-uz}e^{-u}}{\overline S(t)}=(1+z)e^{-(1+z)u}\quad(u\geq0).
$$
Hence
$$
\boxed{U\mid T>t\sim\operatorname{Exp}(1+\theta t^2/2),\qquad \mathbb E[U\mid T>t]=\frac1{1+\theta t^2/2}.}
$$
The <conditional expectation> starts at one and declines toward zero when $\theta>0$. Survival favors the less frail individuals, precisely explaining why the <population hazard under exponential frailty> equals $h_0(t)$ times a decreasing multiplier. The original PDF asks for conditioning on the event $T>t$; the TeX's misplaced inequality is a transcription error, not conditioning on an observed exact event time.
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