Approximations and Derivatives of Probability Functions

Abstract

A very important tool in reliability based systems design and optimization are, see [1],[6],[9], probability functions of the type

$$ P\left( x \right): = P\left( {{y_{li}} < \left( \leqslant \right){y_i}\left( {a\left( \omega \right),x} \right) < \left( \leqslant \right){y_{2i}},i = l, \ldots ,m} \right),x \in {\mathbb{R}^n} $$

(1)

$$ {P_f}\left( x \right): = P\left( {\mathop {\min }\limits_{1im} {\text{ }}{g_i}\left( {a\left( \omega \right),x} \right) < o} \right),x \in {\mathbb{R}^n} $$

(2)