Stat 5101 Lecture Notes - School of Statistics

2.7. INDEPENDENCE 772.7 Independence2.7.1 Two DefinitionsLindgren (p. 79, equation (3)) gives the following as a definition of independentrandom variables.Definition 2.7.1 (Independent Random Variables).Random variables X and Y are independent ifP (X ∈ A and Y ∈ B) =P(X∈A)P(Y ∈B). (2.57)for every event A in the range of X and B in the range of Y .We take a quite different statement as the definition.Definition 2.7.2 (Independent Random Variables).Random variables X and Y are independent ifE{g(X)h(Y )} = E{g(X)}E{h(Y )} (2.58)for all real-valued functions g and h such that these expectations exist.These two definitions are equivalent—meaning they define the same concept.That means that we could take either statement as the definition and prove theother. Lindgren takes (2.57) as the definition and “proves” (2.58). This isTheorem 11 of Chapter 4 in Lindgren. But the “proof” contains a lot of handwaving. A correct proof is beyond the scope of this course.That’s one reason why we take Definition 2.7.2 as the definition of the concept.Then Definition 2.7.1 describes the trivial special case of Definition 2.7.2in which the functions in question are indicator functions, that is, (2.57) saysexactly the same thing asE{I A (X)I B (Y )} = E{I A (X)}E{I B (Y )}. (2.59)only in different notation. Thus if we take Definition 2.7.2 as the definition, weeasily (trivially) prove (2.57). But the other way around, the proof is beyondthe scope of this course.2.7.2 The Factorization CriterionTheorem 2.46 (Factorization Criterion). A finite set of real-valued randomvariables is independent if and only if their joint distribution is the product ofthe marginals.What this says is that X 1 , ..., X n are independent if and only iff X1,...,X n(x 1 ,...,x n )=n∏f Xi (x i ) (2.60)i=1