Differentiation of probability functions: The transformation method

In reliability-oriented design and optimization of engineering systems, one needs various derivatives of the probability of systems survival P(x)=P(Yl y(a(w),x) Yu) Here, y = y(a,x) denotes the vector of response or output variables depending on the design or input vector x and the vector of random system parameters a = a(ω); we assume that a(ω) has a given probability density function = (a). Furthermore, yℓ, yu are the vectors of given lower and upper bounds for y. There is shown that in many cases derivatives of arbitrary order can be obtained by applying an integral transformation Tx to the integral representation of P(x) such that the transformed domain of integration becomes independent of x. The derivatives result then by interchanging differentiation and integration. Based on the mean value representations of found in the first part, estimations of the derivatives can be obtained by using several sampling techniques. Furthermore, having the mentioned integral representations, can be computed approximately by writing first as a Laplace integral and applying then the asymptotic expansion techniques known for Laplace integrals.