Using Finitely Additive Probability: Uniform Distributions on the Natural Numbers
作者:
JosephB. Kadane,
Anthony O'Hagan,
期刊:
Journal of the American Statistical Association
(Taylor Available online 1995)
卷期:
Volume 90,
issue 430
页码: 626-631
ISSN:0162-1459
年代: 1995
DOI:10.1080/01621459.1995.10476556
出版商: Taylor & Francis Group
关键词: Charges;Extension of finitely additive probabilities;Number theory;Prime numbers;Probability bounds;Residue classes
数据来源: Taylor
摘要:
In the usual, countably additive definition of probability, it is not possible to have a distribution giving equal probabilities to every one of the natural numbers. Yet such a distribution would be interesting and potentially useful. This article considers an approach to this problem based on finitely additive probability. We give a necessary and sufficient condition for when specifications of the probabilities of an arbitrary collection of subsets of a space Ω can be extended to define a finitely additive probability on all the subsets of Ω. This is applied to probability statements modeling the uniform distribution on the natural numbers, using relative frequencies and residue classes to make precise notions of uniformity. Tight bounds are given on the possible values of the probability of an arbitrary set under both interpretations. These bounds are applied to several sets of interest.
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