How Broad is the Class of Normal Scale Mixtures?

Abstract

We study the class of scale mixtures of normal distributions with mean zero. Given that the cdf F(x) of such a mixture is fixed at two points, say F(x_1) = \alpha_1, F(x_2) = \alpha_2, we answer the question of how widely F(x_3) can vary at some third point x_3. A brief final section mentions extensions of our theorem. 1.