Mixtures are not derivable

Abstract

A binary operation o on probability distribution functions is derivable from a binary operation on random variables if there exists a two-place functionV such that, for any distribution functionsF andG, there exist random variablesX andY, defined on a common probability space, such thatF andG are the distribution functions ofX andY , respectively, and o(F, G) is the distribution function ofV (X, Y). We show that if o(F, G) =cF + (1 -c)G, 0 <c < 1, then o is not derivable; similarly,\(\phi (F,G) = \sqrt {FG} \) is not derivable.