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.
A: As far as I know, this statement makes sense in two contexts--as the definition of an event or as a constraint on a variable--but is a little imprecise.
Marco, Procrastinator, and Michael Chernick have discussed the "event" part above, but a concrete example might help. Suppose you have two random variables $X$ and $Y$ which are determined by rolling a pair of fair dice. You could be interested in the event ${X>Y}$ i.e., how often the number on the first die is strictly larger than the number on the second. In this particular case $P(X > Y)=15/36$, which is easy enough (in this case) to get by enumerating all the possibilities.
It also makes sense when defining random variables. I can't think of a good example off the top of my head, but imagine that $X$ is a person's age and $Y$ is how long they've had some disease. Then, obviously, we know that $Y \le X$. 
That said, in general, I'm not sure it makes a ton of sense to compare two arbitrary random variables, at least not without defining exactly what you mean by less-than, greater-to, and equal up front. In a casual context, I can imagine someone using $X > Y$ to mean $\mathbb{E}(X) > \mathbb{E}(Y)$. It would be better to be more explicit about it, since it could also  refer to other things (e.g., median, upper bound, etc). Plus, keep in mind that some of these quantities aren't defined for all random variables: for example, a Cauchy-distributed random variable doesn't have a mean.
Finally (and pedantically), random variables often have units. If $X$ is the balance of my retirement account and $Y$ is the number of viral particles in a sample, it doesn't make a whole lot of sense to talk about $X > Y$, or even $\mathbb{E}(X) > \mathbb{E}(Y)$.
A: Ordering random variables is a thing, e.g., the concept of stochastic dominance. Basically, the ordering is based on the n-th order moments and/or cumulative distributional function (CDF) of a random variable. However, the ordering is partial, not total, i.e., you cannot induce an ordering into the entire space of random variables; in other words, some variables are not comparable. 
