TABLE OF CONTENTS In Chapter 2 it was seen that events are independent if the probability of a joint event can be written as a product of probabilities of individual events. The notion of independent events provides a corresponding notion of independent random variables and, as will be seen, results in random variables being independent if their joint distributions are product distributions.
Two random variables X and Y defined on a probability space are inde pendent if the events X ?1( F) and Y ?1( G) are independent for all F and G in 
( 
). A collection of random variables { X i, i = 0, 1, , k ? 1} is said to be independent or mutually independent if all collections of events of the form 
are mutually independent for any F i ? ( ); i = 0, 1, , k ?1.
Thus two random variables are independent if and only if their output events correspond to independent input events. Translating this statement into distributions yields the following.
Random variables X and Y are independent if and only if
Recall that F 1 F 2 is an alternative notation for 
we will frequently use the alternative notation when the number of product events is small. Note that a product and not an intersection is used here. The reader...
