(To appear in The American Statistician.) Distance covariance (Sz\'ekely, Rizzo, and Bakirov, 2007) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables $X$ and $Y$. This approach deserves to be touched upon in modern courses on mathematical statistics. It makes use of distances of the type $|X-X'|$ and $|Y-Y'|$, where $(X',Y')$ is an independent copy of $(X,Y)$. This raises natural questions about independence of variables like $X-X'$ and $Y-Y'$, about the connection between Cov$(|X-X'|,|Y-Y'|)$ and the covariance between doubly centered distances, and about necessary and sufficient conditions for independence. We show some basic results and present a new and nontechnical counterexample to a common fallacy, which provides more insight. We also show some motivating examples involving bivariate distributions and contingency tables, which can be used as didactic material for introducing distance correlation.

翻译：（即将发表于《美国统计学家》）距离协方差（Székely, Rizzo, and Bakirov, 2007）是一个引人入胜的新近概念，作为检验随机变量$X$与$Y$之间任意类型依赖关系的工具而广受欢迎。这一方法值得在现代数理统计课程中予以探讨。它利用了$|X-X'|$和$|Y-Y'|$类型的距离，其中$(X',Y')$是$(X,Y)$的独立副本。这自然引出了关于$X-X'$与$Y-Y'$等变量独立性、Cov$(|X-X'|,|Y-Y'|)$与双重中心化距离协方差之间的关联，以及关于独立性的充分必要条件的若干问题。我们展示了一些基本结果，并针对一个常见谬误提出了一个新的非技术性反例，从而提供了更深入的理解。我们还展示了一些涉及二元分布和列联表的启发性示例，这些材料可作为引入距离相关性的教学素材。