Differentiation formulas for probability functions: The transformation method

The reliability of a technical or an economical stochastic system is measured usually by probability functions of the type P ( x ) = P ( y ~ . , < ( ~ ) y i ( a ( o , ) , x) < ( , < ) y , , , i = l . . . . . m) (1) a n d / o r P i ( x ) = P i ( z . . i < ( < ~ ) y i ( a ( c o ) , x ) < (~<)y,,i) , i = 1 . . . . . m. ( l . l ) Here y ~ , , < ( ~ ) y , ( a , x ) < ( V ) y , , , , i = l . . . . . , , , (2) are the basic operating conditions or behavioral constraints of the underlying system, where Y = ( Yl . . . . . Yi . . . . . Y,,)' (3) are certain response, output or performance variables, e.g. displacements, stresses, strains, forces, weight in structural design or the total costs, deviations between the demands and production outputs in a production problem. The response variables y/ are functions y i = Y i ( a , x) , i = 1 . . . . . rn, (3.1) of a decision r-vector x = ( x I . . . . . x k , . . . , x r ) ' and a parameter u-vector a = (a I . . . . . aj . . . . . a~)', where xe, k = 1 . . . . . r, are the determinist ic (nomina l ) design input variables or deterministic system coefficients, e.g. sizing variables, geometrical