Generalized transformations of random variables

We present a formula for all the densities that a random variable may have so that a given transform of it has a given density. The collection of densities which are changed by a prescribed transform to have a prescribed density may be called the generalized transformations of a random variable. This terminology is analogous to 'generalized inverse' for reasons given in the paper. Special cases include theorems of Roberts and Geisser: (i) the densities that a random variable may have if its mth absolute power is gamma (Roberts, 1971), (ii) the densities that a random variable may have if its square is gamma (Roberts and Geisser, 1966).