算术拔河与本福德定律

摘要：

本福德定律预测，在现实生活中，数字最左边的第一个有效数字在所有可能的1到9位数字之间的比例近似为1 + 1/digit，因此低位数字比高位数字出现的频率要高得多。关于遵守本福德定律的数据配置的两个基本先决条件是高数量级和正偏度，尾部落在直方图的右侧，因此数量配置是这样的，小的是众多的，大的是罕见的。在本福德定律的研究中，一个相关的主题是随机变量的乘法和加法之间的鲜明对比，以及它们不同的结果数量和数字配置。随机增殖过程引起数量级的显著增加，并且它们倾向于偏对数正态分布，有利于小的而不是大的。另一方面，随机添加过程不会引起数量级的任何增加，并且它们倾向于由中心极限定理预测的对称正态分布，有利于中等而不是小和大。因此，虽然乘法过程对本福德行为是高度有益的，但是加法过程对本福德行为是高度有害的。在这篇文章中，它表明，经常在现实生活中的数据，乘法和加法过程混合在一起，在一个测量或表达式，因此，他们激烈竞争的优势，每一个试图施加最大的影响大小和数字。加法和乘法之间这场拉锯战的胜负取决于生成的随机变量的数量级，也取决于交战双方的相对实力，这一点可以用该过程的代数式中所涉及的比较算术来衡量。

英文摘要：

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. The two essential prerequisites for data configuration with regards to compliance with Benford's Law are high order of magnitude and positive skewness with a tail falling to the right of the histogram, so that quantitative configuration is such that the small is numerous and the big is rare. A related topic in the study of Benford's Law is the stark contrast between multiplications and additions of random variables and their distinct resultant quantitative and digital configurations. Random multiplication processes induce substantial increase in order of magnitude and they tend to the skewed Lognormal Distribution, favoring the small over the big. Random addition processes on the other hand do not induce any increase in order of magnitude and they tend to the symmetrical Normal Distribution as predicated by the Central Limit Theorem, favoring the medium over the small and the big. Thus, while multiplication processes are highly conducive to Benford behavior, addition processes are highly detrimental to Benford behavior. In this article it is shown that often in real-life data, multiplication and addition processes mix together within one measurement or expression, and consequently they fiercely compete for dominance, each attempting to exert the greatest influence upon sizes and digits. Such tugs of war between additions and multiplications are won or lost depending on the orders of magnitude of the generating random variables, as well as on the relative strength of the two warring sides, measured in terms of the comparative arithmetical involvement in the algebraic expression of the process.

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