An Application of Numerical Integration Techniques to Statistical Tolerancing, III—General Distributions

Suppose one wants numerical estimates for the moments of the distribution of a response X which is a function of the statistically independent random variables yk , k = 1, 2, …, n, when the yk are from known distributions. The functional relationship between X and the yk is assumed to be known in the generalized sense that for given values of the yk the response X may be obtained somehow, e.g., by experiment., by engineering calculations, by analog. The same problem was considered in the first paper in this series (Evans, 1967) for the special case in which the yk were all from normal distributions; here the distributions for the yk are general. A quadrature formula is developed for approximating the moments of X within an error O(σ5) where σ2 is a representative variance for the yk . It uses the values for X obtained for 2n 2 + 1 selected arguments, (y 1, y 2, …, yn ). Efficient quadrature formulas are given for the case in which multiple distributions of the yk are of interest. A numerical example is in...