7.1 Applicability of the Stress-Strength Model

In previous chapters we have discussed in some detail the following two main topics:

1. Expressions for the probability P(X and its generalizations for various distributions of X and Y;

2. Expressions and properties of various estimators of P(X and its generalizations based on a random sample as well as other sampling procedures.

We have seen that these topics often involve challenging calculations and are of great usefulness when viewed from probabilistic and statistical aspects. We have also attempted to describe in the earlier chapters the genesis and motivation for probabilities of the type P(X and their connection with the classical non-parametric tests of equality of two distribution functions based on the extensively used and popular Wilcoxon-Mann-Whitney statistic.

It would seem that Birnbaum (1956) was perhaps one of the first researchers who dealt with the model P(X in stress-strength content. It is worth quoting Birnbaum s Illustration as presented in his pioneering paper which may serve as the road-map of the research in the last 45 years. His ideas are in resonance with the observations by Bilikam (1985) which appeared in engineering literature some 30 years later:

An illustration. If structural components of a mechanism are mass produced, the strength at failure Y of each single component (equals stress at which this component will fail) may be considered a random variable. The component is installed in an assembly and exposed to a stress which reaches its maximum value X,