Just to make sure I get exponential distributions correct. Would this be correct
"If $\displaystyle X$ follows a Poission distribution, then $\displaystyle T$ measures the probabality that you have to wait $\displaystyle t_a$ time periods until $\displaystyle X \geq 1$."

(Where $\displaystyle X \geq 1$ is the same things as saying "$\displaystyle X$ har occured", for example "the component is dead.")