We consider the one-dimensional Poisson process. Suppose for some a > 0 it is given that there are exactly two points in [0, a], or in other words: Na = 2. The goal of this exercise is to determine...

We consider the one-dimensional Poisson process. Suppose for some a > 0 it is given that there are exactly two points in [0, a], or in other words: Na = 2. The goal of this exercise is to determine the joint distribution of X1 and X2, the locations of the two points, conditional on Na = 2.

a. Prove that for 0

b. Deduce from a that