Hello, Given a probability distribution of a discrete variable p1(x) and a probability distribution of a discrete variable p2(y) defined by p2(y) = Sum_{x,x'} p1(x) p1(x') * KroneckerDelta((x+x')/2 = y). (1) Let F1(x) be the cumulative distribution function (CDF) of p1(x): F1(x) == Sum_{x'<=x} p1(x') and let F2(x) be the CDF p2(x).

Is there a way of expressing F2(x) ONLY in terms of F1(x)? If there is none, is there any known (tight) upper and lower bound for F2(x) that is a function ONLY of F1(x)?

Thank you! Best Michele