Finding pdf of $X$ given joint pdf $f(x,y)=\frac{1}{4}(y-x)e^{-y}1_{-y<x<y,0<y<\infty}$

Question

So I have to find the density function of $X$ for the below function with $C = \frac{1}{4}$ $$f(x,y)=C(y-x)e^{-y},\qquad-y\lt x\lt y \quad \text{and} \quad 0 < y < \infty$$

and the solution in the book is:

So based on the comment in the book it looks like you have to find the integral for $x > 0$ and $x < 0$. Why is calculating the integral using $(0, \infty)$ wrong? Or even calculating the integral of $(0, \infty)$ for $x > 0$ and $(-\infty, 0)$ for $x < 0$ wrong?

Answer 1

The domain of the joint density is the area above $|x|$ (red).

So the marginal distribution is given by the following integral for any $x$:

$$f_X(x)=\int_{|x|}^{\infty}\frac14 (y-x)e^{-y}\ d y=\frac14e^{-|x|}(|x|-x+1).$$