# Poisson process

• Oct 21st 2009, 11:22 AM
Joolz
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!!
• Oct 21st 2009, 12:32 PM
Laurent
Quote:

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 $T$ 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 $S_4+S_5$ where $S_4,S_5$ are independent exponential r.v. of parameter $\lambda$ (respectively representing the time spent in states 3 and 4).