On classical occupancy problems II. (Sequential Occupancy)

Békéssy, András (1964) On classical occupancy problems II. (Sequential Occupancy). A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (1-2). pp. 133-141.

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Abstract

We consider, as in Part I [1], a random distribution of balls in n cells, assuming that the balls are randomly and independently dropped into the cells with the same probability 1/n. In Part I the number of the balls was taken to be fixed, and the "state" of the set of cells was regarded a random variable. Suppose now that certain parameters, characterising a "state" of cells, are fixed, while the random variable is the number of balls necessary to reach that given state. Let be к the number of cells, which contain less than m + 1 (m = 0, 1, 2, . . .) balls, and let be v(n, m, k) the number of independent throws. Most results concern the random variable v(n, 0, k) i.e. the number of balls needed to obtain at least one ball in each, except k cells (k = 0, 1, 2, . . .). Probability distributions, moments and limiting distributions related to v(n, 0 , k) have been determined [ 2 ], [ 3 ]. D. J. Newmann and L. Sherp and later on P . Erdős and A. Rényi have dealt with the expectation and with the limiting distribution of v(n, m, 0) [4], [5].¹ In the present paper two theorems on the limiting distribution of v(n, m, k) are given.

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