I have a system consisting of components $S_1$ and $S_2$ whose lifetimes $T_1$ and $T_2$ follow the exponential distribution with parameter $\lambda$. At time $t=0$ the component $S_1$ is switched on and $S_2$ is kept off until $S_1$ fails (and is immediately switched on). What is the distribution of the lifetime of the system?
To me the logical solution would be $f(T_1,\lambda)+f(T_2,\lambda)$ where f is the probability density function $f(x,\lambda)=\lambda e^{-\lambda x}$.
$\Rightarrow$ $\lambda e^{-\lambda T_1}+\lambda e^{-\lambda T_2}$
but I have nothing to verify it with. Am I on the right track?