Mixtures are not derivable

AbstractA binary operation ø 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 ø(F, G) is the distribution function ofV (X, Y). We show that if ø(F, G) =cF + (1 -c)G, 0 <c < 1, then ø is not derivable; similarly, $$\phi (F,G) = \sqrt {FG} $$ is not derivable.