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Similar to cumulative distribution function\u00a0(CDF), it uniquely defines the random variable and can be evaluated with a simple one-dimensional minimization formula. This work leverages the structure of that formula to introduce buffered CDF (bCDF)\u00a0and reduced CDF (rCDF)\u00a0for random vectors. bCDF and rCDF are shown to be the minimal Schur-convex upper bound and the maximal Schur-concave lower bound of the multivariate CDF, respectively. Special structure of bCDF and rCDF is used to construct an algorithm for solving optimization problems with bCDF and rCDF in objective or constraints. 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