Two-emitter photobleaching completion time
= Two-emitter photobleaching completion time
{title2=$f(t)=2k(e^{-kt}-e^{-2kt})$}
Two initially active independent emitters with exponential lifetimes of rate $k$ have last-bleaching time $T=\max(T_1,T_2)$. Independence gives $P(T\leq t)=(1-e^{-kt})^2$ for $t\geq0$. Differentiating proves the density in the title, and integrating its survival function gives $E[T]=3/(2k)$. The first event has rate $2k$, while the waiting time from first to second has rate $k$.