Suppose a set of $m$ integers from $0$ to $n-1$. The integers are uniformly distributed and unique in the set ($n \gg m$). Then, put all the integers into a list an sort that list: $x_0 < x_1 < \dots{} < x_{m-1}$
What is the probability distribution of $x_i$, and of $d_i := x_{i+1} - x_i$ ?
I stumbled on a very similar question, but it only discusses the minimal and average value of $d_i$. It does however present an interesting way to think of it:
Give a rope of size $n$, I cut it on $m$ places chosen uniformly at random (such that, in my case, no two cuts are done at the same spot). What is the probability distribution of the position of the $i^{th}$ cut ? What is the probability distribution of the size of the rope bits ?
I ran some simulations, and $d_i$ seems to have an exponential, maybe Gamma distribution (which is intuitive), and $x_i$ is bell-shaped (probably hypergeometric, which again is rather intuitive) but I cannot figure it out on paper. Any thoughts ?