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Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Yl, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.

Consider the clustering statistic, k* defined to be the maximum number of 1's to appear within any m consecutive positions in a random arrangement of a 1's and M–a 0's. Pr(k* – k) was found for the case k ≧ ½ a by the author [4], and then for m/M = 1/L, L an integer, by Naus [3]. Here, we generalise Naus's method to obtain the distribution of k* under no special restrictions on a, k, m or M.

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On sequential occupancy problems

Note on a clustering problem

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2 results

On sequential occupancy problems

Note on a clustering problem