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This article presents strategies for evaluating the mean, variance, and failure probability of a response variable given measurements subject to both epistemic and aleatory uncertainties. We focus on a case in which standard sensor calibration techniques cannot be used to eliminate measurement error since the uncertainties affecting the metrology system depend upon the measurement itself (e.g., the sensor bias is not constant and the measurement noise is colored). To this end, we first characterize all possible realizations of the true response that might have led to each of such measurements. This process yields a surrogate of the data for the unobservable true response taking the form of a random variable. Each of these variables, called a Random Datum Model (RDM), is manufactured according to a measurement and to the underlying structure of the uncertainty.