Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class Cf of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to Cf: this happens, for instance, when f(x) = Ce−λx or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.
