Q) Let runners arrive on an infinitely long track with a mean density of 1 per meter. No two runners stand at the same place and the number of runners standing in any two disjoint segments are independent of each other. All the runners run, independently, an exponentially distributed random distance with mean $20m$. Prove that the resulting runner locations has the same distribution as before.
I know that the number of runners in a length $l$, follow $\sim Pois(l)$ distribution and each runner is given a mark(i.e. the distance they run) and that a marked Poisson process is also a Poisson process but not sure how to translate "resulting runner locations has the same distribution as before" into an equation? Is it the joint distribution of the runner locations?