= Solution
In a <frailty model>, an individual's unobserved susceptibility to an event is represented by a nonnegative <frailty random variable> multiplying that individual's <hazard function>. Conditional on its value, subjects with the same measured covariates can therefore have different event rates. The frailty is a persistent latent characteristic of the individual, not newly sampled noise at each time.
Here the conditional <hazard function> is $u e^{\beta z}h_0(t)$, with <baseline hazard> $h_0$. A multiplier $u>1$ raises susceptibility and $u<1$ lowers it. Normalization $\mathbb E U=1$ fixes the frailty scale: without it, multiplying every frailty by a constant and dividing the <baseline hazard> by that constant would give the same individual hazards. The <variance> of $U$ measures unobserved heterogeneity. An atom at $U=0$, if allowed, would give a permanently zero conditional hazard.
The <hazard ratio> between groups is $e^\beta$ for individuals compared at the same frailty. It need not equal the population <hazard ratio>, because <survival selection> changes the frailty distribution among those still alive or event-free.
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