I have two independent, exponential RVs $X$ and $Y$ that both have the same parameter. I am trying to find the distribution of $Y$ given that $X>Y$. So far, I have:
$$P(Y|X>Y) = P(Y=y|X>Y) = P(Y=y, X>Y)/P(X>Y)$$
and I don't know how to proceed at this point. I understand how to get the denominator, but the numerator is really confusing me. Could someone give me a hint?