TITLE: Median $\neq$ expectation
QUESTION [6 upvotes]: Do you have an example of real random variable such that its median is remarkably different from its expectation? I'd like an example where it is obvious that they are different.

REPLY [3 votes]: The Cauchy distribution, with pdf $\frac 1 \pi \frac{1}{1+x^2}$, has no expectation but its median is $0$.