Macroscopic analogs of quantum-mechanical phenomena and auto-transformations of functions

The two main goals of the present article are: 1) To prove an existence theorem for forbidden zones for the expectations of real-valued random variables. 2) To define transformations (named here as auto-transformations) of the probability density functions (PDFs) of random variables into similar PDFs having smaller sizes of their domains and to outline their basic features. Such transformations can be used also for functions beyond the scope of the probability theory. The goals are caused by the well-known problems of behavioral sciences, e.g., by the underweighting of high and the overweighting of low probabilities, risk aversion, the Allais paradox, etc.