# Expected value of biggest distance of adjacent points uniformly picked in $[0,1]$

We pick $$n\ge 2$$ points in $$[0,1]$$ with uniform distribution. What is the expected value of the largest distance of $$2$$ adjacent points?

• Isn't this closely related to the expected value of the biggest of $n$ numbers picked uniformly from $[0,1]$? – quarague Jul 3 '19 at 7:42

The expected value is asymptotic to $$(\log n)/n$$ as $$n$$ tends to infinity (By "asymptotic" I mean that the ratio tends to 1). One way to see this is to use the representation of order statistics of uniform points as the first $$n$$ points of a Poisson process, normalized by the $$n+1$$ Point. Since the sum of $$n+1$$ exponential variables is concentrated, the question reduces to the distribution of the maximum of $$n-1$$ Exponential variables.
Usually one considers the maximum $$M_n$$ of the $$n+1$$ gaps including the gap between zero and the first point and between 1 and the last point, but that does not change the asymptotics. The known properties of $$M_n$$ are surveyed in the introduction of Devroye's paper that Brendan McKay mentioned: https://projecteuclid.org/download/pdf_1/euclid.aop/1176994313