Decreasing population hazard under constant individual hazards (${\overline{h}}^{\prime }(t)=-\mathrm{Var}(\Lambda \mid T>t)\le 0$)

For a nonnegative random constant individual hazard $\Lambda$, write $M_k(t)=\mathbb{E}[{\Lambda }^k{e}^{-\Lambda t}]$. The population survivor function is $M_0$ and its hazard is $M_1/M_0$. Under the integrability needed to differentiate, $M_k^{\prime }=-M_{k+1}$, soThe conditional distribution is weighted by $e^{-\Lambda t}$. This proves decreasing population hazard without any decrease in individual hazards and explains survival selection in a heterogeneous population.