Finding the probability of a joint density

Question

Given the joint density of two random variables $X$ and $Y$,

$f_{XY}(x,y)=2e^{-(x+y)}$ for $0<x<y$

How do I compute $P(Y<1|X=1)$?

I know the conditional probability formula is:

$P(Y<1|X=1)=\frac{P(X=1,Y<1)}{P(X=1)}=\frac{\int\int2e^{-(x+y)}dydx}{\int\int2e^{-(x+y)}dydx}$

However, I'm unsure about the bounds of the integral

Answer 1

The PDF tells us that $X<Y$ a.s. so that $P(Y<1\mid X=1)=0$.