This work introduces and characterizes the so called "random extended intervals", these are random intervals for which the left bound may be higher than the right one. To carry out this study, we introduce on the set of random extended intervals a structure of metric space relevant to study extended interval-valued ARMA time series. This is done by extending the Hausdorff metric on extended intervals and defining a family of metrics dγ relevant for the set of random extended intervals and which do not have some disadvantages of the Hausdorff metric. We show that there exists a unique metric dγ for which γ(t)dt is what we have called "adapted measure" and we use this metric to define variability for random extended intervals.