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$X$ 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!

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    A somewhat related question is "how different can the median be from the mean?" The answer is [at most one standard deviation](http://en.wikipedia.org/wiki/An_inequality_on_location_and_scale_parameters#An_application:_distance_between_the_mean_and_the_median).2011-11-18

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