Probabilities for a $k$th Nearest Neighbor Problem on the Line
Sylvan R. Wallenstein, Joseph Naus · The Annals of Probability · 1973
Given $N$ points distributed at random on $\lbrack 0, 1)$, let $p_n$ be the size of the smallest interval that contains $n$ points. Previous work finds $\operatorname{Pr}(p_n \leqq p)$, for $n > N/2$, and for $n \leqq N/2, p = 1/L, L$ an integer. This paper finds the distribution of $p_n$, for all $n, N,$ and $p$.