Abstract

A number of standard textbooks that are followed in a junior/senior level course or in a first-year graduate level course in mathematical statistics and probability, routinely include one single basic illustration, obviously in its variant forms, to highlight an important point: pairwise independence may not imply (mutual) independence. We earnestly believe that beginning students appreciate more examples to clarify these key issues. Hence, we hope that our new sets of nontrivial illustrations from Section 2 will help our audience. Next, in Section 3, we extend the notion to q-wise independence with a large set of illustrations using both discrete and continuous random variables showing that q-wise independence may not imply (mutual) independence. We reasonably assure that this discourse is immediately accessible to juniors/seniors and first-year graduate students.

摘要

在初级/高级课程或研究生一年级数学统计和概率课程中遵循的许多标准教科书通常包含一个单一的基本插图，显然以不同的形式，以突出一个重要点：成对独立可能并不意味着（相互）独立。我们真诚地相信，初学者会喜欢更多的例子来阐明这些关键问题。因此，我们希望第 2 节中的一组新的重要插图能够帮助我们的观众。接下来，在第 3 节中，我们将概念扩展到q - wise 独立性，并使用大量插图，使用离散和连续随机变量表明q明智的独立可能并不意味着（相互）独立。我们有理由保证，大三/大四和一年级研究生可以立即访问此课程。