I have to find the marginal density $f_y(y)$ of this function:
$$f_{X,Y}(x,y)=\tfrac 18(x^2 - y^2)e^{-x}\quad\Big[ x \in( 0 ;\infty), y \in (-x; x)\Big]$$
However, I'm confused about which borders I have to use for the integration. Should it be integral from 0 to infinity or from x to infinity because of the domain of y?
Thank you!
Well, just rearrange the construction so that the bounds of $y$ are free from $x$, and the bounds of $x$ are in terms of $y$. $$\begin{align}\{(x,y): x \in( 0 ;\infty), y \in (-x; x)\} ~&=~ \{(x,y):
0 So$$f_Y(y) = \mathbf 1_{y\in(-\infty;\infty)}\cdot \int_{\bbox[0.25ex,border:dotted 1pt green]?}^{\bbox[0.25ex,border:dotted 1pt green]?} \tfrac 18(x^2 - y^2)e^{-x}\operatorname d x$$