Bassino et al.~2010 and Regnier et al.~1998 showed the generating functions of the distributions of the number of the occurrences of words (distributions of words for short) in finite string in the form of rational functions. However the coefficients of the expansion of the rational functions are complicated and we do not have a simple formula of the exact distributions of words from rational functions. In this paper we study the finite dimensional generating functions of the distribution of nonoverlapping words for each fixed sample size and show the explicit formula of the distributions of words for Bernoulli model. We demonstrate that 1) the tests based on the distributions of words reject the random number generator in BSD Library with p-value almost zero and 2) computation of the distributions of words in the human DNA size strings.

翻译：Bassino et al.~2010 和 Regnier et al. ~1998 显示了以理性函数的形式在有限字符串中出现的单词数(短词数分配)的分布功能,但是,理性函数扩展的系数是复杂的,我们没有从理性函数中准确分配单词的简单公式。在本文中,我们研究了为每个固定样本大小分配非重叠单词的有限维度生成函数,并展示了Bernoulli 模型单词分布的清晰公式。我们证明,1)基于单词分布的测试拒绝BSD 库的随机数字生成器,P值几乎为零,2)计算人类DNA大小字符的分布。