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Approximations and derivatives of probability functions. (English) Zbl 0854.60020
Anastassiou, George (ed.) et al., Approximation, probability, and related fields. Proceedings of a conference, Santa Barbara, CA, USA, May 20-22, 1993. New York, NY: Plenum. 367-377 (1994).
In reliability-oriented design and optimization of engineering systems one needs derivatives of the probability of systems survival \(P(x) = P(y_a < y(a (\omega), x) < y_b)\) of various orders. Here, \(y = y(a,x)\) denotes the vector of response variables of a structural system depending on the design vector \(x\) and the vector of random system parameters \(a = a (\omega)\) having a given probability density function \(\varphi = \varphi (a)\); furthermore, \(y_a\), \(y_b\) are the vectors of given lower and upper bounds for \(y\). For a large class of functions \(y = y(a,x)\) arising e.g. in optimal structural design there is shown that derivatives of \(P = P(x)\) of arbitrary order can be obtained by applying an appropriate integral transformation to the integral representation of \(P(x)\): The derivatives of \(P(x)\) result then by interchanging differentiation and integration. In some cases the transformation has to be applied in combination with a certain stochastic completion technique and a final limit process. The derivatives are represented again by integrals or expectations. The differentiation formula is applied to determine approximations of \(P = P(x)\) based on approximations of the response function \(y = y(a, x)\).
For the entire collection see [Zbl 0840.00040].

60E99 Distribution theory
60K10 Applications of renewal theory (reliability, demand theory, etc.)
74P99 Optimization problems in solid mechanics