First moment inequality implies tail distribution inequality? Let $U,V$ be two continuous random variables, both with continuous CDF. Suppose that $\mathbb E V \geq \mathbb E U$. Can one conclude that $\mathbb P(V> x) \geq \mathbb P(U>x)$ for all $x\geq 0$? If not, what additional conditions are needed?

Here is an argument with may make think it's not true: if $Y$ is a positive random variable, then $E(X)=\int_0^{+\infty}P(X>t)dt$. The fact that $P(U>x)\geqslant P(V>x)$ seems much stronger than $E(U)\geqslant E(V)$.
