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Suppose we have a Poisson process of parameter $\lambda$. Each event of this Poisson process represents a start date of a period which duration is a random variable that follows an exponential distribution with parameter $\mu$.

Let pick a period $T$ among these periods. $T$ follows an exponential distribution. Let $N$ be a random variable representing the number of events (theses events include those of the Poisson process, and events related to the end dates of the periods) during the period $T$.

I would like to know how to determine the probability distribution of $N$ and its Expected value.

Thank you.

watou
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  • Is picking the period $T$ done randomly? – Gregory Grant Jun 11 '15 at 17:25
  • @GregoryGrant yes. – watou Jun 11 '15 at 18:24
  • Well the distribution $N$ follows a poisson given $T$, so it's a poisson with random parameter. Is it possible that $N$ is poisson with its parameter given by an exponential? – Gregory Grant Jun 11 '15 at 18:46
  • Does this help? http://math.stackexchange.com/questions/281013/poisson-distribution-with-exponential-parameter – Gregory Grant Jun 11 '15 at 18:47
  • @GregoryGrant How can you be sure that $N$ follows a poisson distribution? There are two type of events: Events of the poisson process, and events corresponding to end dates of periods. – watou Jun 11 '15 at 19:11
  • The sum of two independent Poisson processes is again Poisson, so if the process counting events corresponding to end dates of the periods is also Poisson (and independent of the first Poisson process), then $N$ is Poisson. – Mankind Jun 13 '15 at 15:40
  • @HowDoIMath But, the second process is counting events corresponding to end dates of the periods and the periods follow exponential distribution with parameter $\mu$. – watou Jun 14 '15 at 14:20
  • @GregoryGrant, The link you gave me doesn't really help me. Because we have two types of events: those of the Poisson process and those corresponding to end dates (which do not follow a Poisson process). – watou Jul 02 '15 at 18:46

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