Note on y-truncation: A simple approach to generating bounded distributions for environmental applications

It may sometimes be desirable to introduce bounds into probability distributions to formalise the presence of upper or lower physical limits to data to which the distribution has been applied. For example, an upper bound in raindrop sizes might be represented by introducing an upper bound to an exponential drop-size distribution. However, the standard method of truncating unbounded probability distributions yields distributions with non-zero probability density at the resulting bounds. In reality it is likely that physical bounding processes in nature increase in intensity as the bound is approached, causing a progressive decline in observation relative frequency to zero at the bound. Truncation below a y-axis point is proposed as a simple alternative means of creating more natural truncated probability distributions for application to data of this type. The resulting “y-truncated” distributions have similarities with the traditional truncated distributions but probability densities have the desirable feature of always declining to zero at the bounds. In addition, the y-truncation approach can also serve in its own right as a means of creating a rich new class of bounded probability distributions when transformations of y-truncated distributions are included.