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I know that to determine whether $X$ and $Y$ are independent I have to find the marginal distributions of $X$ and $Y$. I've already found the marginal probability density function $f_Y$, but I'm stuck on integrating the joint pdf with respect to $y$ to get the marginal probability density function $f_X$. I've already entered it into integral calculators and none of them were able to find the integral. Any help would be appreciated!

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There is an easier way: $$f_Y(y)=Cye^{-y}\int_0^\infty e^{-x/y}dx=Cy^2e^{-y}$$ And, $$f_{X|Y=y}(x)=\frac{f_{X,Y}(x,y)}{f_Y(y)}=\frac{Cye^{-y-x/y}}{Cy^2e^{-y}}=\frac{1}{y}e^{-x/y}$$

The term $y$ hasn't vanished in the conditional density of $X$; so $f_{X|Y=y}(x)\neq f_X(x)$ in general, which means they're not independent.

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There is an even easier way here. If $X$ and $Y$ are independent then the joint density kernel will be seperable, meaning that you can split it as:

$$f(x,y) \propto g(x) h(y).$$

In the present case you have a term $e^{-x/y}$ in the density, and there is no way to seperate this term into a product of functions of $x$ and $y$. Therefore your random variables are not independent.

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