This article deals with Stein characterizations of probability distributions. We provide a general framework for interpreting these in terms of the parameters of the underlying distribution. In order to do so we introduce two concepts (a class of functions and an operator) which generalize those...

It is proved that the average length of standard confidence intervals for parameters of Gamma and normal distributions monotonically decreases with the sample size. The proofs are based on fine properties of the classical Gamma function.