# Determining the Expected value of a random variable

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.

• 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? math.stackexchange.com/questions/281013/… – 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