Determination of the parameters of the Rosin-Rammler and beta distributions from their mode and variance using equation-solving software

The mode and variance of Rosin-Rammler (Weibull) and beta distributions can be calculated directly from their corresponding function parameters. Determination of the function parameters from the mode and variance, however, can only be done numerically. Modern software enables us to do this, through simultaneous solution of the necessary equations. Thus, changes in distribution characteristics can be conveniently expressed in terms of the corresponding changes in mode and variance and be readily transformed into the parameters of the distribution. In this way, not only do the changes become intuitively clearer but they can also be implemented in simulations of an evolving size distribution pattern during size reduction or enlargement processes, where the mode decreases or increases monotonically, while the variance reaches a maximum at an intermediate size. Although the demonstrations shown were generated with only the Rosin-Rammler and beta functions, the programs and methods employed can also be applied to other distribution functions.