Good, so X and X are random variables. Let me put it in this way:
1. If X is a random variable, then X is also a random variable too.
2. If X is a random variable, then X is a random variable too.
First case, we say that X and X are two different random variables. But I do not see any condition for X to be a random variable given that X is a random variable, so I think this statement is not true at all.
Second, since we can not map back a real number to the sample space, say using X, then the fact that X is a random variable does not imply that X is a random variable too, so this statement is false.
What do you think?
