A random variable is defined as a function from a probability space $\Omega$ to $\mathbb{R}$ (with certain properties). I think I understand the notion of independence, but my question is:
If two variables $X$ and $Y$ have the same probability distribution, aren't they the same mathematical object (that is, the same function)?
Then, why can $X$ be independent from $Y$ but not from $X$?
What does it mean to have two "different" random variables?
Again, I understand this from a practical point of view, but I am not sure about the formalization.