On the physical and probabilistic consistency of some engineering random models

• Examples of random variables used in civil engineering models are provided.
• Consistency of some important mathematical models used in civil engineering are analyzed.
• Dimensional problems of some models are identified and solved by providing alternatives.
• How an extreme value analysis has to be performed is clarified.
• Multivariate random models are classified as under, over or uniquely determined.
• Rules are given in order to build models without the reported inconsistencies.

In this paper we deal with the probability and physical consistency of random variables and models used in engineering design. We analyze and discuss the conditions for a model to be consistent from two different points of view: probabilistic and physical (dimensional analysis). The first leads us to the concept of probabilistically consistent models, which arises when the joint distributions of all variables are required. This implies that relations among the variables must be respected by densities and resulting moments. In particular the most common linear, product and quotient relations, which are physically justified, must be especially considered. Similarly, stability with respect to minimum or maximum operations and consistency with respect to extremes (maxima and minima) arises in practice. From the dimensional analysis point of view, some models are demonstrated to be inconsistent. In particular, log-normal and chi-squared models are shown to be non adequate for location or location-scale variables. The problem of building compatible models based on conditional distributions and regression functions is analyzed too. It is shown that incompatible models can be easily obtained if a consistency analysis is not performed. All these and other problems are discussed and some models in the literature are analyzed from these two points of view. When some families fail to satisfy the desired properties, alternative models are provided. Finally, some simple examples and conclusions are given to summarize the analysis.