Let \(U_1\) and \(U_2\) be independent and uniformly distributed on \([0,1]\). Consider their order statistics \(U_{(1)} < U_{(2)}\); what is their joint pdf?
Prove that for a Poisson point process on \([0, \infty)\), constructed with exponential inter-arrival times, that conditional that there exactly two points in \([0,1]\), then those points are independently and uniformly distributed.
Demonstrate this fact by simulations. Think about what you need to illustrate with the simulation.