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A natural way of quantifying the degree of linear dependence between two fuzzy random variables in certain models is analyzed. In these models, the linear relationship is formalized in terms of a regression model based on the usual arithmetic for fuzzy sets. The degree of linear relationship is proposed to be measured by means of a kind of determination coefficient, that is, through the proportion of variability of the response fuzzy random variable explained by the regression model. Some properties of both the models and the determination coefficient are analyzed in order to verify the suitability of this coefficient as a measure of linear dependence. Finally, the meaning of a correlation coefficient which mimics the usual one for real-valued random variables is discussed.