Some Probabilities, Expectations and Variances for the Size of Largest Clusters and Smallest Intervals

Given N points independently drawn from the uniform distribution on (0, 1), let ᵱ_n be the size of the smallest interval that contains n out of the N points; let ñ_p be the largest number of points to be found in any subinterval of (0, 1) of length p. This paper uses a result of Karlin, McGregor, Barton, and Mallows to determine the distribution of ñ_p, for p = 1/k, k an integer. The paper gives simple determinations for the expectations and variances of ᵱ_n, for all fixed n > (N + 1)/2, and of ñ_1/2. The distribution and expectation of ñ_p are estimated and tabulated for the cases p = 0.1(0.1)0.9, N = 2(1)10.