# Thread: Poisson process

1. ## Poisson process

Ok, here is my question

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!!

2. Originally Posted by Joolz
Ok, here is my question

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!!
The definition of $\displaystyle T$ is not very clear but I guess you understood correctly what they mean. Then it looks correct. So you need to find the distribution of $\displaystyle S_4+S_5$ where $\displaystyle S_4,S_5$ are independent exponential r.v. of parameter $\displaystyle \lambda$ (respectively representing the time spent in states 3 and 4).