Abstract The idea of transforming one random variate to another with a more convenient density has been developed in the first half of the 20th century. In his thesis, Norman L. Johnson (1917–2004) developed a pioneering system of transformations of the standard normal distribution which gained substantial popularity in the second half of the 20th century and beyond. In Johnson’s 1949 Biometrika paper entitled Systems of frequency curves generated by methods of translation, summarizing that thesis, one of his primary interests was the behavior of the shape of the probability density functions as their parameter values change. Herein, we attempt to further elucidate this behavior through a series of geometric expositions of that transformation process. In these expositions insight is obtained into the behavior of Johnson’s density functions, and their skewness and kurtosis, as they converge to their limiting distributions, a topic which received little attention.

中文翻译：

---

约翰逊频率曲线系统——历史、图形和限制性观点

摘要 在 20 世纪上半叶，提出了将一个随机变量转换为另一个具有更方便密度的随机变量的想法。在他的论文中，Norman L. Johnson (1917-2004) 开发了一个开创性的标准正态分布变换系统，该系统在 20 世纪下半叶及以后大受欢迎。在 Johnson 1949 年的 Biometrika 论文中，题为“由转换方法生成的频率曲线系统”的论文总结了该论文，他的主要兴趣之一是概率密度函数的形状随着参数值的变化而变化。在此，我们试图通过对该变换过程的一系列几何说明来进一步阐明这种行为。在这些说明中，可以深入了解约翰逊密度函数的行为，