Independent classes of random variables  (Page 5/6)

Introduction

The concept of independence for classes of events is developed in terms of a product rule. In this unit, we extend the concept to classes of random variables.

Independent pairs

Recall that for a random variable X , the inverse image $X^{-1}\left(M\right)$ (i.e., the set of all outcomes $\omega \in \Omega$ which are mapped into M by X ) is an event for each reasonable subset M on the real line. Similarly, the inverse image $Y^{-1}\left(N\right)$ is an event determined by random variable Y for each reasonable set N . We extend the notion of independence to a pair of random variables by requiring independence of the events they determine. More precisely,

Definition

A pair $\{X,Y\}$ of random variables is (stochastically) independent iff each pair of events $\{X^{-1}\left(M\right),Y^{-1}\left(N\right)\}$ is independent.

This condition may be stated in terms of the product rule

Independence implies

Note that the product rule on the distribution function is equivalent to the condition the product rule holds for the inverse images of a special class of sets $\{M,N\}$ of the form $M=(-\infty ,t]$ and $N=(-\infty ,u]$ . An important theorem from measure theory ensures that if the product rule holds for this special classit holds for the general class of $\{M,N\}$ . Thus we may assert

The pair $\{X,Y\}$ is independent iff the following product rule holds

If there is a joint density function, then the relationship to the joint distribution function makes it clear that the pair is independent iff the product rule holds for the density. That is, thepair is independent iff