# How are two random variables equal to one another?

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 $X_1 = X_2$?

• This question points to the weakness of the standard definition of a random variable solely as a function from measurable sets to real numbers. In the sense that they are solely such a function, you can say $X=Y$, and you don't need to add a.e. But no one would say that $X=Y$ unless you are almost surely guaranteed the same outcome. – Paul Feb 22 '16 at 16:39
• In the answer to your edited question, no, you cannot make that jump unless you know something specific about the covariances. – Paul Feb 22 '16 at 16:41
• Paul. I do actually have knowledge of the covariances. What am I supposed to use and how? – Simp Feb 22 '16 at 16:43
• If the correlation is 1, you do know that they are equal. Otherwise, you cannot assume that. – Paul Feb 22 '16 at 16:45
• Yes, of course. If the correlation is $1$, then we know that $X_1 = \alpha + \beta X_2$ and, since they have the same distribution $\alpha = 0, \beta = 1$. However, the theorem I have at hand establishes the equality only almost everywhere. Will this influence by desired conclusion? Will I have to change my conclusion so that it says "the joint distribution lies on the diagonal $\textbf{with probability 1}$"? – Simp Feb 22 '16 at 16:52

If I saw $X=Y$ I would probably assume it means either literal equality of functions or equality almost surely.