The standard method to check for the independence of two real-valued random variables -- demonstrating that the bivariate joint distribution factors into the product of its marginals -- is both necessary and sufficient. Here we present a simple necessary condition based on the support sets of the random variables, which -- if not satisfied -- avoids the need to extract the marginals from the joint in demonstrating dependence. We review, in an accessible manner, the measure-theoretic, topological, and probabilistic details necessary to establish the background for the old and new ideas presented here. We prove our result in both the discrete case (where the basic ideas emerge in a simple setting), the continuous case (where serious complications emerge), and for general real-valued random variables, and we illustrate the use of our condition in three simple examples.
翻译:检查两个实际估价随机变数的独立性的标准方法 -- -- 表明其边缘产品中的双变量联合分布系数是必要和充分的。我们在此根据随机变数的支持组合提出一个简单的必要条件,如果不能满足,则无需从组合中提取边际以证明依赖性。我们以方便的方式审查为确定此处介绍的旧思想和新思想的背景所必需的测量理论、地貌学和概率细节。我们证明了我们的结果,既包括离散情况(基本思想在简单环境中出现)、连续情况(出现严重并发症),也包括一般实际估价随机变数,我们用三个简单的例子来说明我们状况的使用情况。