  The increased uncertainty and complexity of nonlinear systems have motivated
investigators to consider generalized approaches to defining an entropy
function. New insights are achieved by defining the average uncertainty in the
probability domain as a transformation of entropy functions. The Shannon
entropy when transformed to the probability domain is the weighted geometric
mean of the probabilities. For the exponential and Gaussian distributions, we
show that the weighted geometric mean of the distribution is equal to the
density of the distribution at the location plus the scale, i.e. at the width
of the distribution. The average uncertainty is generalized via the weighted
generalized mean, in which the moment is a function of the nonlinear source.
Both the Renyi and Tsallis entropies transform to this definition of the
generalized average uncertainty in the probability domain. For the generalized
Pareto and Student's t-distributions, which are the maximum entropy
distributions for these generalized entropies, the appropriate weighted
generalized mean also equals the density of the distribution at the location
plus scale. A coupled entropy function is proposed, which is equal to the
normalized Tsallis entropy divided by one plus the coupling.
