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On a model for a fair distribution of gifts

Published online by Cambridge University Press:  14 July 2016

János Galambos
János Galambos*
Affiliation:
Temple University, Philadelphia
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Extract

A number of items are to be distributed among n individuals, arranged in a line and labelled by the integers 1, 2, ···, n, in such a way that the distance between any two members presented should be at least d, a given number, but otherwise any subset of the population should have the same probability of each of its members being presented. The number of items thus distributed is a random variable X(d, n) and it turns out that X(d, n) is asymptotically a fixed percentage of the population size n. This fact makes it possible to apply the model in a number of situations. For example, if in a foreign aid program a number of tools are offered and to be distributed in a fair way but, for lack of sufficient quantity, one tool is expected to be used by a number of neighbours, the model is applicable if the amount of aid is not fixed in advance but may vary within certain limits. A similar situation arises in cases of disaster (floods, earthquakes, etc.) when the victims are sent gifts, again in an insufficient quantity, hence for the sake of justice, a family of a smaller size is supposed to receive only one gift. In many other similar cases, the model can be used.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 681 - 690
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Daykin, D. E. (1960) Representation of natural numbers as sums of generalized Fibonacci numbers. J. London Math. Soc. 35, 143–160.Google Scholar
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[3] Galambos, J. (1970) The occurrence of a renewal process in the theory of numbers. Abstract. Ann. Math. Statist. 41, 1784.Google Scholar
[4] Galambos, J. (1971) A generalization of a theorem of Borel concerning the distribution of digits in dyadic expansions. Amer. Math. Monthly, 78 (To appear).Google Scholar