We are given a homogeneous Poisson point process with parameter $m$ on the non-negative half-line. Suppose we know there are exactly $k$ points in the interval $[0,T]$. Pick a point $a$ uniformly at random from these $k$ points.
I want to show that the arrival time $T_a$ of $a$ is uniformly distributed in the interval $[0,T]$.
In some course notes I found a link between Poisson processes and the uniform distribution, but I can't make it work as I have little experience in probability theory.
Any help would be greatly appreciated!
PS: This is not homework - I'm trying to understand Proposition 20 in this paper.