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I want to calculate the marginal distribution of $X$ given that the joint probability density function of $(X,Y)$ is given by $$f(x,y)=2592(x^2-y^2)e^{-2x} \qquad 0<x<\infty,\ -x<y<x$$

My problem is to determine the bounds of integration for the marginal density. Are they going to be from $-x$ to $0$ and then from $0$ to $x$? Or just from $0$ to $x$ since $x$ can only assume positive values?

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    – whuber ♦
    Oct 24, 2020 at 15:46

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The support of $Y$ conditional on the value of $X=x$ is $(-x,x)$ so that should be your region of integration.

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  • $\begingroup$ Thank you very much for your reply I appreciate it. However, after I submitted this question I thought of this again and I computed the integral from -x to 0 and add this to the integral from 0 to x. Is this also true?Or is it just the integral from -x to x correct? I am sorry for asking so many questions but i am having some difficulties. Thank you for your help! $\endgroup$
    – Thekla
    Oct 23, 2020 at 6:57
  • $\begingroup$ The integral is additive, so yes it's the same thing $\endgroup$
    – dmh
    Oct 23, 2020 at 13:41

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