If we generate 10 uniform samples in sorted order and take the difference between the 9th and 10th samples?

Question

I'm wondering what's going on here, if we consider an experiment where we generate X uniform samples from a distribution, and then take the difference between two consecutive samples from that set of samples, let's say the 9th and 10th. Now if we repeat this process N times, we can get a distribution for the difference between the 10th and 9th sample.

The resulting distribution is what? Why is it tailed to the left as well? Can someone help me explain this?

Answer 1

Based on comments:

If the distribution is uniform, then you can consider taking a circle and $11$ points on it, taking one as the beginning and end (the $0$th and $11$th) and the others as your ten samples. Symmetry/exchangeability will tell you all $11$ gaps have the same distribution, and it is particularly easy to find the distribution function and density of the first gap. It should be intuitively clear that small gaps are more likely than large gaps, since a small gap gives more possibilities for the other gaps than a large gap does.

If the distribution is uniform on $[0,1]$ you in effect have a uniform Dirichlet distribution for the $11$ gaps (including below the minimum and above the maximum of the sample). For each gap, the marginal CDF is $1−(1−x)^{10}$ and so the marginal density $10(1−x)^9$ when $0≤x≤1$.

This is a Beta distribution with parameters $\alpha=1$ and $\beta=10$. With a larger sample and suitable rescaling, the distribution converges towards an exponential distribution, though this is only an approximation for a finite sample: note $10(1−x)^9\approx 10e^{−10x}$.

You said in a comment "I just tried doing this sampling using R and it seems that you get a Normal distribution". That is not what should happen and the following simulation suggests my result is correct, with small values more likely than large values and a close to exponential distribution. It takes $10^5$ simulations of sample size $10$ and looks at the gap between the $9$th and $10$th values, both as a CDF and as a density. (You could look at all the gaps by plotting sims rather than sims[10,] but you would get the same sort of thing.)

Simulated data is in black, theoretical curves in red (essentially overlapping the simulated data) and approximate exponential curves in blue:

set.seed(2022)
gaps <- function(samplesize){
  diff(sort(c(0, runif(samplesize), 1)))
  }   
sims <- replicate(100000, gaps(10))
plot.ecdf(sims[10,])
curve(1-(1-x)^10, add=TRUE, col="red")
curve(1-exp(-11*x), add=TRUE, col="blue")

plot(density(sims[10,]))
curve(10*(1-x)^9, add=TRUE, col="red")
curve(10*exp(-10*x), add=TRUE, col="blue")