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Math Help - Poisson process

  1. #1
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    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!!
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Joolz View Post
    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).
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