Some Unifying Concepts in Reliablity Physics, Mathematical Models, and Statistics

The basic concepts of dimensional analysis are used to make a general examination of reliability physics, mathematical models, and statistics. Beginning with reliability physics, an examination of Eyring\´s equation is made using Buckingham\´s Pi theorem. The Arrhenius equation is then seen to be an approximation to the Eyring equation. The use of dimensional analysis in constructing mathematical models based on empirical investigations is considered. The results suggest that, in every empirical study, the appropriate dimensionless variables should first be identified. These variables should then be used in experimental exploration and for subsequent mathematical modeling. The probability density functions of statistics may be associated with patterns of "noise". The exponential, Weibull, and normal distributions serve as examples. In this context, statistical methods are used for extracting signals from a pattern of noise and for judging the statistical significance of the signal-to-noise ratio. The pattern of noise is important in deriving an appropriate method of signal extraction. Examples of signal-to-noise ratios are given by the common tests of statistical significance. The u-test for the significance of the difference between an observed mean and its hypothesized value and the t-test of significance for the difference between two observed means are typical examples. The maximization of the signal-to-noise ratio can be used to obtain important classification methods. The loinear discriminant, used in some reliability screening applications, furnishes an example of such a maximization.