Pairwise Independence May Not Imply Independence: New Illustrations and a Generalization

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.