关于大数定律的简单注解

Discussion on the Law of Large Numbers

本文从“抛硬币实验”出发,激发对频率和概率关系的思考,得出随着试验次数的增加频率逐渐趋于概率的结论。这个结论被数学家伯努利用与抛硬币相类似的缶子模型所证明,后称为伯努利弱大数定律。而伯努利时代遗留下来的问题,试验次数无限时概率值可否用频率值替代,直到20世纪初才得以解决。波莱尔证明了其正确性,结论更强的大数定律由此诞生。随后,本文就更一般的强弱大数定律分别从直观意义和测度意义上展开了讨论。最后,基于经验分布函数给出了伯努利场合下强大数定律的应用。

In this paper,the relation between frequency and probability is discussed by using the“coin experiment”.The conclusion drawn is that the value of frequency approaches the probability as the number of experiment increases.This was proved by James Bernoulli using a model similar to the“coin experiment”and known as Bernoulli’s Weak Law of Large Numbers later.However,there was a problem left:whether frequency and probability are equal under the infinite condition.It was not until the 1900’s that the problem was proved to be correct.Subsequently,Strong Law of Numbers and Weak Law of Numbers are compared in a more general condition in terms of intuition and measurement.And finally,application of the Strong Law of Numbers in Bernoulli case is given based on the empirical distribution function.