If $X$ and $Y$ are two random variables with the exact same distribution, do we then say $$X = Y?$$
Or do we say $$X = Y \ \text{almost everywhere}?$$
And if so, why? $X$ and $Y$ are maps. Why are tiwo maps equal just because they happen to have the same distribution?
And if not, when do we say they are equal?
EDIT with context:
I have two marginal distributions ($X_1, X_2)$ which are normally distributed with same mean and variance. I need to show that their joint distribution lies in a 1-dimensional space in $\mathbb{R}^2$, since it is a singular normal distribution. I would like to conclude that $X_1 = X_2$ and thus the 1D space is just the diagonal. But can I go from "$X_1$ and $X_2$ have the same distribution to saying $X_1 = X_2$?
Best Answer
Having the same distribution is very weak, much weaker than equality almost surely. In fact it does not even depend on the random variables being defined on the same probability space.
If I saw $X=Y$ I would probably assume it means either literal equality of functions or equality almost surely.