Formulas for risk of failure

The concept of stress strength inference plays an important role in modeling problems with respect to engineering fracture mechanics, see, e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] and [21]. The risk of failure of a system can be expressed by the probability R = Pr(X2 < X1), where X1 and X2 are some random variables representing the stress and the strength experienced by the system. In this note, we derive a comprehensive collection of formulas for R by assuming the most commonly known models for X1 and X2. We feel that this work could serve as a useful reference for engineering fracture mechanics.