Construction and transformation of probability densities *

Abstract

We present a simple unified approach for the transformation of probability densities. The approach relates any two probability densities by an operator transformation using the standard differentiation operator. We obtain a generalization of the Gram-Charlier and Edgeworth series. We further generalize the approach by showing that the differentiation operator can be replaced by an arbitrary Hermitian operator. As a further special case we obtain the operator transformation that scales the mean and standard deviation of a probability density. Furthermore, we show how the unusual formalism of quantum mechanics for constructing densities can be related to standard probability theory. We do so by generalising the theorem of Khinchin regarding characteristic functions.