The original problem is:
3 workers work on 3 tasks, each require time distributed exp(1) independently. The next task will begin at time T when two workers finished their task. Calculate E(T).
Answer by book: E(T) = 5:6
I understand that "T" is the middle of the 3 work times, but how to find its PDF? thanks in advance
For $t\geq 0$ let $N\left(t\right)$ denote the number of workers that finished their job in interval after $t$.
Then $N\left(t\right)$ has binomial distribution with parameters $n=3$ and $p=e^{-t}$.
Further we have:$$T>t\iff N\left(t\right)\geq2$$ so we conclude that: $$P\left(T>t\right)=P\left(N\left(t\right)\geq2\right)=3e^{-2t}\left(1-e^{-t}\right)+e^{-3t}=3e^{-2t}-2e^{-3t}$$
I leave the rest to you.