$X$ is a continuous random variable (we can assume some statistic (e.g., mean and variance) are known, but the distribution is unknown). Consider a probability $\operatorname{Pr}(X<\operatorname{E}(X))$.
We know for symmetric distributions, $\operatorname{Pr}(X\leq\operatorname{E}(X))=0.5$. However, for asymmetric distributions, is there any approximation to approximate this probability? is there an upper or a lower bound expression for this probability?
Thanks!