Joint Probability of $f_{1,2}(x,y) = e^-yI(0<x<y<\infty)$

Question

Let Joint Probability Density Function of Random variables X,Y be

$$f_{1,2}(x,y) = e^{-y}\mathcal I_{(0<x<y<\infty)}$$

then I want to find the numerical value of $P(X+Y \le 1)$

So I had tried my integration as below:

$$\int_0^1\int_0^{1-x}e^{-y}\mathrm dy\mathrm dx$$

However, this value doesn't match to the answer but cannot figure out why.

Any correction or hint which I am missing?

Answer 1

Note that $I(0<x<y<\infty)$ tells us that we have $x < y$ with probability $1$. Note also that $1-x > x \implies x < 1/2$. Your integral should therefore look like $$ \int_0^{1/2} \int_x^{1-x} e^{-y}\,dy\,dx $$