This work sheds some light on the relationship between a distribution's standard deviation and its range, a topic that has been discussed extensively in the literature. While many previous studies have proposed inequalities or relationships that depend on the shape of the population distribution, the approach here is built on a family of bounded probability distributions based on skewing functions. We offer closed-form expressions for its moments and the asymptotic behavior as the support's semi-range tends to zero and $\infty$. We also establish an inequality in which the well-known Popoviciu's one is a special case. Finally, we provide an example using US dollar prices in four different currencies traded on foreign exchange markets to illustrate the results developed here.
翻译:这项工作揭示了分配标准偏差及其范围之间的关系,文献中已经广泛讨论了这个专题。虽然许多先前的研究都提出了取决于人口分布形态的不平等或关系,但这里的方法建立在基于扭曲功能的受约束概率分布的大家庭上。我们用封闭式的表达方式表达其时刻和无药可救的行为,因为支持的半程为零和美元。我们还建立了一种不平等,众所周知的波波维基乌(Popoviciu)是一个特殊的例子。最后,我们举一个例子,用在外汇市场上交易的四种不同货币的美元价格来说明在这里产生的结果。</s>