  There exist several simple representations of uncertainty that are easier to
handle than more general ones. Among them are random sets, possibility
distributions, probability intervals, and more recently Ferson's p-boxes and
Neumaier's clouds. Both for theoretical and practical considerations, it is
very useful to know whether one representation is equivalent to or can be
approximated by other ones. In this paper, we define a generalized form of
usual p-boxes. These generalized p-boxes have interesting connections with
other previously known representations. In particular, we show that they are
equivalent to pairs of possibility distributions, and that they are special
kinds of random sets. They are also the missing link between p-boxes and
clouds, which are the topic of the second part of this study.
