Let there be some initial stock (continuum) of naked people $n$. Clothes arrive at rate $\lambda$. At the same time, there is an inflow of naked people at the same rate - such that we have $\dot n = 0$.
Denote the length at which a person was naked $d$. How can I compute $Prob(d \in (a, b])$?
I'm familiar with the Poisson distribution, and have read into Poisson point processes. From there, I get that the probability of an event arriving at rate $\lambda$ not happening between $(a, b]$ is given by $\exp(-\lambda(b - a))$. How do I go on from here? This probability is naturally the same, independent of waiting duration - I'd need to compute the implied distribution over duration, and then divide measure of people at some interval over the total mass?