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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!!

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It looks like you have to consider two cases:

  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 $\int\limits_{x}^{\infty}f(x,y)dy$

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  • $\begingroup$ 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 :/ $\endgroup$
    – nicefella
    Commented Apr 13, 2013 at 18:40
  • $\begingroup$ 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 $x<0$. $\endgroup$
    – Matt L.
    Commented Apr 13, 2013 at 18:43
  • $\begingroup$ Ah I see! Thank you for your help :) $\endgroup$
    – nicefella
    Commented Apr 13, 2013 at 19:05

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