Let (Nt)t>=0 be a Poisson process with parameter lambda. S is the time spent in state 2. T is the time after S, that it takes for the process to be in state 5.

1) What is the probability distribution of S?

2) What is the probability distribution of T?

My ideas:

1) S is a holding time, thus S ~ Exp(lambda)

2) Say S=S3 then T= S4 + S5 where S4 ~ Exp(lambda) and S5 ~ Exp(lambda)

Can someone tell me if I'm thinking in the right direction?

Thanks a lot!!