Inequality of two independent random variables My question is related to this one but more specific. Inequality on two random variables
We have two continuous random variables, $X$ and $Y$. We know that the expected value of both is 0 (or more generally, they have equal expected values). We know that they are independent. But we are not sure if they are identically distributed. Can we say that $P(X>Y)=P(X<Y)=0.5$? 
In case, the above information is not adequate to claim $P(X>Y)=P(X<Y)=0.5$:
I performed a sampling experiment in R to test this. I sampled 1000 values of $X$ and $Y$ each from normal distribution, with means 0 but different values of standard deviation. The proportion of observations such that $X>Y$ was indeed $\sim$0.5. So, I am tempted to say that if independence and equality of means is not adequate to make the claim, symmetrical distributions should be adequate. Do we really need the two variables to be identically distributed? Is independence and equality of means sufficient for $P(X>Y)=P(X<Y)=0.5$? 
 A: In simple words, for continuous random variables that are independent and symmetrically distributed around 0, $P(X=x, Y=y)=P(X=-x, Y=-y)$. Hence, $X>Y$ and $X<Y$ should be equally likely. Importantly, $X$ and $Y$ do not have to be identically distributed.

Assuming that the sampling distributions for $X$ and $Y$ are symmetrical about 0. $P(X>Y)$ can be expressed in terms of $P(\left| X \right| > \left| Y \right|)$ and $P(\left| X \right| < \left| Y \right|)$:
$P(X>Y) = P(\left| X \right| > \left| Y \right|)P(X>0 | \left| X \right| > \left| Y \right|) + P(\left| X \right| < \left| Y \right|)P(Y<0 | \left| X \right| < \left| Y \right|)$.
But $P(X>0 | \left| X \right| > \left| Y \right|) = P(X>0)=0.5$ and similarly, $P(Y<0 | \left| X \right| < \left| Y \right|) = P(Y<0)=0.5$. Hence,
$P(X>Y) = 0.5\left[P(\left| X \right| > \left| Y \right|) + P(\left| X \right| < \left| Y \right|)\right]$.
Similarly,
$P(X<Y) = P(\left| X \right| < \left| Y \right|)P(Y>0 | \left| X \right| < \left| Y \right|) + P(\left| X \right| > \left| Y \right|)P(X<0 | \left| X \right| > \left| Y \right|)$. 
Again, $P(X<0 | \left| X \right| > \left| Y \right|) = P(X<0)=0.5$ and $P(Y>0 | \left| X \right| < \left| Y \right|) = P(Y>0)=0.5$. Hence, we get,
$P(X<Y) = 0.5\left[P(\left| X \right| > \left| Y \right|) + P(\left| X \right| < \left| Y \right|)\right]$.
To summarize, $P(X>Y) = P(X<Y)$ and since $P(X=0)$ for continuous variables, they are both equal to 0.5.
