Fitting a second-order parametric distribution conditioned on an explanatory variable using maximum likelihood estimation

We show how to use the method of maximum likelihood estimation (MLE) to fit a second-order parametric distribution conditioned on a single explanatory variable to data. To illustrate the method, we demonstrate how a second-order log-normal distribution, conditioned on the population served, can model the variability and the parametric uncertainty in data collected by the US Environmental Protection Agency for the concentration of radon 222 in drinking water supplied from ground water, even though 28% of the data fall at or below the minimum reporting level.