Differentiability of probability function

Basic results on differentiability of probability functions are reviewed and systematically presented. Strict mathematical statements are formulated, some basic theorems are informally proved and illustrated with examples. Probability function, from formal point of view, is an expectation of an indicator function or an integral over the domain depending upon the parameter. Gradient of probability function is represented in different forms (integral over the surface, volume, or sum of surface and volume integrals). These results can be used for sensitivity and uncertainty analysis of stochastic models, reliability analysis, Probabilistic Risk Analysis, optimization of the stochastic systems, and other applications involving uncertainties in parameters.

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