Abstract
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 $p$.
Citation
Sylvan R. Wallenstein. Joseph I. Naus. "Probabilities for a $k$th Nearest Neighbor Problem on the Line." Ann. Probab. 1 (1) 188 - 190, February, 1973. https://doi.org/10.1214/aop/1176997037
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