What is the difference between identically distributed random variables ?

Question

I know that two random variables X and Y are identical distributed, iff $$ P(X \leq x) = P(Y \leq x) $$ for all $x \in \mathbb{R}$. Doesn't that mean, that $$ P(X = x) = P(Y = x) $$ for all $x \in \mathbb{R}$? I can't see, where they could differ then. Where does $$ P(X \leq x) = P(Y \leq x) \Rightarrow P(X = x) = P(Y = x) \Rightarrow X=Y $$ go wrong ? A counterexample would be perfect.

Answer 1

It is true that $$P(X=x) = P(Y=x)$$ But that does not imply that the variables are the same.

You and I toss a coin; call my toss $X$ and yours $Y$. Then $$P(X=head) = P(Y=head) = 1/2$$ But that does not mean that if I get head, then you will get head too, i.e. $X \neq Y$.