Ok, so here is the question: Suppose $X$ has density $f$ and $\mathbb E|X|<\infty$. Prove: $m$ is a median of $X$ iff the function $g(a)=\mathbb E(|X-a|)$ is minimized at $a=m$. Recall $m$ is a median of $X$ iff $P(X \leq m)\geq 1/2$ and $P(X \geq m)\geq 1/2$.
For discrete case, I think I can just rearrange the data as order statistics. However, for continuous case, I tried to manipulte the integrals of the expectation but didn't achieve the conclusion.
