= Decreasing population hazard under constant individual hazards
{title2=$\overline h'(t)=-\operatorname{Var}(\Lambda\mid T>t)\leq0$}
For a nonnegative random constant individual hazard $\Lambda$, write $M_k(t)=\mathbb E[\Lambda^ke^{-\Lambda t}]$. The population survivor function is $M_0$ and its hazard is $M_1/M_0$. Under the integrability needed to differentiate, $M_k'=-M_{k+1}$, so
$$
\overline h'=\frac{-M_2M_0+M_1^2}{M_0^2}=-\operatorname{Var}(\Lambda\mid T>t).
$$
The 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>.
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