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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!

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It depends entirely on the context. The indicator $\mathcal{I}\{X>Y\}$ is a random variable that has a Bernoulli distribution with success probability $p = P(X>Y)$. So, $X>Y$ gives you a single realization from that distribution - a sample of such values will give you information about $p$ - whether or not this is meaningful depends on the context - can you provide more info? – Macro May 26 '12 at 5:20
As Macro says, you can calculate $p={\mathbb P}(X<Y)$, which is known as stress-strength model or in terms of estimators as Pitman's measure of closeness. The keypoint here is, how do you want to compare them? (1) By their difference: calculate the distribution of $X-Y$, (2) By their ratio: calculate the distribution of the ratio $X/Y$ or $Y/X$, (3) more generally, by a measurable function $f(X,Y)$. – user10525 May 26 '12 at 10:15
What properties do you want $\gt$ to have? Normally, this symbol is reserved for a transitive relation. Now, one can find many transitive relations on RVs merely by comparing some numerical property: for instance, we might say that $X\gt Y$ iff the median of $X$ exceeds the median of $Y$, or iff the first quartile of $X$ exceeds the first quartile of $Y$, etc. Although not nonsense, such relations are not very deep or interesting. What more might you be looking for in the $\gt$ relation? – whuber♦ May 26 '12 at 20:35

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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.

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(+1) As long as you can compare $X$ and $Y$ (i.e. not if they are multidimensional etc.), it has a meaning. – gui11aume May 26 '12 at 17:26
@gu11aume In 1-dimensional Euclidean space there is a natural order to numbers. But there are other spaces that orderings can be defined. Distance functions define a partial ordering because many points have the same distance. But for example under Euclidean distance in R$^2$ either X>Y X=Y or X<Y. So we can define the event [X>Y] based on the distance function. – Michael Chernick May 26 '12 at 17:36
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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 $\mathbb{E}(X) > \mathbb{E}(Y)$.

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