TITLE: Can 2 different random variables have the same CDF?
QUESTION [6 upvotes]: I'm looking for proof that two different random variables can have the same Cumulative Distribution Function; in other words, I'd like to disprove that a CDF uniquely defines a random variable. (Probably a silly question, but I couldn't find anything usable so far).

REPLY [4 votes]: Try $X$ such that $P(X=0)=P(X=1)=\frac12$ and $Y=1-X$. Then $Y\ne X$ in the sense that $P(Y=X)=0$ but the distributions of $X$ and $Y$ coincide.
For continuous random variables, choose $X$ uniform on $(0,1)$ and $Y=1-X$, or $X$ standard normal and $Y=-X$.