Let $A$ and $B$ be independent, positive random variables. Why must $E(\min(A, B)) < \min(E(A), E(B))$, where $\min(X, Y)$ is the minimum of $X$ and $Y$?
I would think the opposite, that $E(A, B) > \min(E(A), E(B))$ because $E(\min(A, B))$ weights all possible values.