Two-emitter photobleaching completion time ($f(t)=2k(e^{-kt}-e^{-2kt})$)

Two initially active independent emitters with exponential lifetimes of rate $k$ have last-bleaching time $T=\max (T_1,T_2)$. Independence gives $P(T\le t)=(1-e^{-kt}{)}^2$ for $t\ge 0$. 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 $k$.