Derivatives of probability functions and integrals over sets given by inequalities
The problem is to find the gradient of probability functions \(F = F(x)\) being represented as integrals over sets \(B = B(x)\) given by a finite number of inequalities, \(f_i (x,y) \leq 0\), \(i = 1, \ldots, m\), where the integrand \(p = p(x,y)\) (probability density) as well as the constraints \(f(x,y) \leq 0\) depend on a certain parameter vector \(x\). The gradient with respect to \(x\) is represented here as the sum of integrals taken over a volume and over a surface. The results are applied to the calculation of the gradient of probability functions arising in chance- constrained programming.
scientific article; zbMATH DE number 773896