How does the expected value of a continuous random variable relate to its arithmetic mean, median, etc. in a non-normal distribution (eg. skew-normal)? I'm interested in any common/interesting distributions (eg. log-normal, simple bi/multimodal distributions, anything else weird and wonderful).
I'm looking mostly for qualitative answers, but any quantitative or formulaic answers are also welcome. I'd particularly like to see any visual representations that make it clearer.
Best Answer
(partially converted from my now-deleted comment above)
The expected value and the arithmetic mean are the exact same thing. The median is related to the mean in a non-trivial way but you can say a few things about their relation:
when a distribution is symmetric, the mean and the median are the same
when a distribution is negatively skewed, the median is usually greater than the mean
when a distribution is positively skewed, the median is usually less than the mean