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In recent years, the possibility theory has been investigated by many authors in the field of mathematics and engineering. A possibility distribution (PD) is, from the mathematical point of view, a generalization of a probability distribution, since it can represent the upper envelope of a family of probability distributions. Given a probability distribution, different probability-possibility transformations can be applied to transform the probability distribution into different PDs. Probability-possibility transformations are useful in metrology whenever statistical data must be dealt with inside the possibility theory, for instance, when they have to be associated with other kinds of uncertain and imprecise data to evaluate measurement uncertainty. This paper shows how uncorrelated and correlated random contributions to uncertainty can be effectively represented, processed, and combined in terms of PDs, by means of an original probability-possibility transformation.