What is the $\mu$ s.t. $$\int_{\mu}^{\infty}1-F(x)dx = \int_{-\infty}^{\mu}F(x)dx?$$
Here $F(x) = P(X\leq x).$
Should $\mu$ be the median of X, i.e. $0.5=F(\mu)$? I think $\mu$ should be the point so that $F(\mu) = 1-F(\mu)$, which is the median of X. But how do I derive it mathematically?