I have read that when dealing with random variables, one often forgets about the underlying probability space (Wikipeida). What is a good example (or a couple) of when one does this? I figure that there are two ways to go about this:
(1) a stand-alone random variable where the probability space is too complicated or
(2) two random variables on rather different probability spaces that have the same distribution, which motivates the study of the distributions themselves.
By a good example, I mean one that would motivate the study of distributions without regard to the underlying probability space. I am not looking for an example that could just so happen be studied with distributions, but an example that shows that it is easier to study the random variable with respect to the distribution rather than the underlying probability space.