Environmental concentrations as ratios of random variables

Human-induced environmental change increasingly threatens the stability of socio-ecological systems. Careful statistical characterization of environmental concentrations is critical to quantify and predict the consequences of such changes on human and ecosystems conditions. However, while concentrations are naturally defined as the ratio between solute mass and solvent volume, they have rarely been treated as such, typically limiting the analysis to familiar distributions generically used for any other environmental variable. To address this gap, we propose a more general framework that leverages their definition explicitly as ratios of random variables. We show that the resulting models accurately describe the behavior of nitrate plus nitrite in US rivers and salt concentration in estuaries in the Everglades by accounting for heavy tails potentially emerging when the water volume fluctuates around low values. Models that preclude the presence of heavy tails and the related high probability of extreme concentrations could significantly undermine the accuracy of diagnostic frameworks and the effectiveness of mitigation interventions, especially for soil contamination characterized by a water volume (i.e., soil moisture) frequently approaching zero.

On the product and ratio of Bessel random variables

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product|XY|and the ratio|X/Y|are derived whenXandYare independent Bessel function random variables. An application of the results is provided by tabulating the associated percentage points.

On the product and ratio of Laplace and Bessel random variables

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product|XY|and the ratio|X/Y|are derived whenXandYare Laplace and Bessel function random variables distributed independently of each other.