Q:

Is that meaningful to compare two random variables?

Currently, I was wondering can we just compare two random variables just like the way we compare two real numbers? Does that make sense? Like for instance, $X$ and $Y$ are two random variables, does $X>Y$ mean something? Or is it plainly nonsense? Anyone has some ideas? Your opinion is greatly appreciated!

A:

I agree with the comments by Procrastinator and Macro but I think your question is clear and has a direct answer without going into the issue of evaluating it in terms of a probability.  If X and Y are random variables with values on the real line or any other space that can be ordered. then {X>Y} has meaning as a measureable event.  So yes it is meaningful.  Under those circumstances there exists a joint probability measure on the set of pairs (x,y) that are values that can be taken on by X and Y.  If this probability measure has a density integrating the joint density over the set of points where X>Y gives the probability that X is greater than Y.  For discrete distributions this is done by summing the probability over all the discrete points where X>Y.

