Necessary condition for equal probability of rank?

Question

Let $X$ and $Y$ be two continuous random variables. Is $\mathbb{E}[X] = \mathbb{E}[Y]$ a necessary condition for having $P(X>Y) = P(X<Y) = 1/2$? Can anyone provide a counter-example?

Answer 1

Note that you can simplify the problem by taking $Y=0$ without loss of generality (by comparing $X-Y$ and $0$). Then, what is happening here is that you ask whether the mean equals the median. This is not true in general. For example take $Z$ as the standard exponential distribution and take $X=Z-1$. Then $$ P(X<0) = P(Z<1) = 1 - e^{-1} \neq \frac{1}{2} $$