= Event type is independent of first time in exponential competing risks
{title2=$f(t,J=A)=(\lambda/s)se^{-st}$}
For independent exponential competing times with rates $\lambda,\gamma$, the observed first time has rate $s=\lambda+\gamma$, while death-type probability is $\lambda/s$. The joint density factors as $\lambda e^{-st}=(\lambda/s)se^{-st}$. Thus the conditional observed time given type still has rate $s$. Independence of type and time generally fails for time-varying hazard ratios.
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