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I'm trying to find the marginal density functions of $X$ and $Y$ from the joint probability density $$ f(x,y)= \begin{cases} xy &, \quad 0<x<2,0<y<2,x+y<2\\ 0 &, \quad \text{otherwise} \end{cases} $$

While I understand $f_X(x)=\int^\infty_{-\infty}f(x,y)dy$ and likewise for $f_Y(y)$, I'm not exactly sure what $x+y<2$ tells us. Does this affect the bounds of integration at all?

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    $\begingroup$ It does effect the region at which you integrate over to get unity - so you will have a triangle which you need to integrate over. $$\int_0^2dy\int_0^{2-y}dx (..)$$ $\endgroup$
    – Chinny84
    Commented May 14, 2021 at 16:15
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    $\begingroup$ ... which suggests to me that the joint pdf integrates to $\frac23$ and needs to be adjusted $\endgroup$
    – Henry
    Commented May 14, 2021 at 16:32

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Integral bounds

It always helps to graph it out.

$f_Y(y)=\displaystyle\int_0^{2-y}xydx$

And

$f_X(x)=\displaystyle\int_0^{2-x}xydy$

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