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Based on Random Set Theory, procedures are presented for bracketing the results of Monte Carlo simulations in two notable cases: (i) the calculation of the entire distribution of the dependent variable; (ii) the calculation of the CDF of a particular value of the dependent variable (e.g. reliability analyses). The presented procedures are not intrusive in that they can be equally applied when the functional relationship between the dependent variable and independent variables is known analytically and when it is a complex computer model (black box). Also, the proposed procedures can handle probabilistic (with any type of input joint PDF), interval-valued, set-valued, and random set-valued input information, as well as any combination thereof.When exact or outer bounds on the function image can be calculated, the bounds on the CDF of the dependent variable guarantee 100% confidence, and allow for an explicit and exact evaluation of the error involved in the calculation of the CDF. These bounds are often enough to make decisions, and require a minimal amount of functional evaluations. A procedure for effectively approximating the CDF of the dependent variable is also proposed.An example shows that, compared to Monte Carlo simulations, the number of functional evaluations is reduced by orders of magnitude and that the convergence rate increases tenfold.