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.