A Quantile-Based Probabilistic Mean Value Theorem

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

Comparison Between Dcrit Considering the Abrupt Variation and Inflexion in the Concrete Mercury Intrusion Porosimetry Curve

Mercury intrusion porosimetry (MIP) has been widely used to evaluate the quality of concrete through the pore size distribution parameters. Two of these parameters are the critical pore diameter (Dcrit) and the percentage of the most interconnected net of pores compared to the total volume of pores. Some researchers consider Dcrit as the diameter obtained from the inflexion point of the cumulative mercury intrusion curve while others consider Dcrit as the diameter obtained from the point of abrupt variation in the same curve. This study aims to analyze two groups of concretes of varying w/c ratios, one cast with pozzolanic cement and another with high initial strength cement, in order to determine which of these diameters feature a better correlation with the quality parameters of the concretes. The concrete quality parameters used for the evaluations were (1) the w/c ratios and (2) chloride diffusion coefficients measured at approximately 90 days. MIP cumulative distributions of the same concretes were also measured at about 90 days, and Dcrit values were determined (1) from the point of abrupt variation and alternatively, (2) from the inflexion point of each of these plots. It was found that Dcrit values measured from the point of abrupt variation were useful indicators of the quality of the concrete, but the Dcrit values based on the inflexion points were not. Hence, it is recommended that Dcrit and the percentage of the most interconnected volume of pores should be obtained considering the point of abrupt variation of the cumulative curve of pore size distribution.