# Limits of Integration for marginal pdf

I just had a small question as something is bothering me. I am trying to find the marginal pdf of the following joint pdf:

$$f(x,y) = (1/8)(y^2 - x^2)e^{-y}$$ where $$-y \le x \le y$$, $$0 < y < \infty$$

My question is this:

For the marginal pdf of $$X$$ we must integrate out with respect to $$Y$$. I understand that the domain $$-y \le x \le y$$ can be re-written like this:

$$-y \le x \le y$$

$$|x| \le y$$

I know that the limits must be in terms of $$x$$ since we want the marginal pdf of $$x$$ but I'm not sure where to go from here. Can anyone please explain? Thanks!!

1. For $x<0$ you need to compute $\int\limits_{-x}^{\infty}f(x,y)dy$
2. For $x\ge 0$ you need to compute $\int\limits_{x}^{\infty}f(x,y)dy$
• How did you get $-x$? The absolute value of $x$ cannot be negative right? The answer that I have read online is that the limit becomes $|x$| to infinity but I'm not sure how they came up with that :/ Commented Apr 13, 2013 at 18:40
• Well, that's exactly it: for negative $x$, $-x$ is positive. And the lower limit $|x|$ is exactly the same as the two cases you must consider, because $|x|=x$ for $x\ge 0$ and $|x|=-x$ for $x<0$. Commented Apr 13, 2013 at 18:43