In this paper, we shall examine the procedure how a random set is realized. We consider three cases. The first case is the ideal case where random set comes as the random α-cut of a given fuzzy set; the second case is that the random set comes as the α- cut of a random fuzzy set for a given cutting level α ϵ [0,1]; the third case is that the random set comes as the random α- cut of a random set. Our study shows that the expected fuzzy set of a random set is strongly affected by the distribution of the random threshold α.