TITLE: Maximum distance between two consecutive points of N random points on a unit length line
QUESTION [7 upvotes]: I have encountered a seemingly simple question on distances of random points.
Place N points randomly and uniformly on the line segment [0..1].
How to derive the expectation (or the distribution) of the "maximum"
distance between two consecutive points?
Just a reference would be very helpful.
Thanks,

REPLY [3 votes]: For $n$ random points on a circle, Theorem 1 of this article provides an expression for the $m$th moment of the maximum distance, namely
$$
\mathbb{E}\left[\left(d_{\max}\right)^m\right]=\frac{(n-1)!}{(n+m-1)!}\sum_{\stackrel{r_1+2r_2+\ldots+m r_m}{r_i\in\mathbb{N}_0}}{\frac{m!}{r_1!1^{r_1}r_2!2^{r_2}\cdots r_m! m^{r_m}}H_{n,1}^{r_1}H_{n,2}^{r_2}\cdots H_{n,m}^{r_m}},
$$
where $H_{n,k}=\sum_{j=1}^n{j^{-k}}$ is the $n$th harmonic number of order $k$.