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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].

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