This is not actually a problem but I need clarification for stochastic ordering From Wikipedia, there is stated: (http://en.wikipedia.org/wiki/Stochastic_ordering) Usual stochastic order: a real random variable A is less than a random variable B in the "usual stochastic order" if Pr(A>x) ≤ Pr(B>x) ,where Pr(°) denotes the probability of an event. Here comes the issue that I do not understand: Characterizations: The following rules describe cases when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist. 1. A ≤ B if and only if for all non-decreasing functions u, E[u(A)] ≤ E[u(B)]. Why it is not possible to attain Pr(A>x) ≥ Pr(B>x) for some x even if E[u(A)] ≤ E[u(B)] ?