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find the marginal distributions of $$ f(x,y) = 2xe^{-y}, \quad 0 < x,\quad x^2 < y $$

I have drawn the support, with $y = x^2$. Not sure how to proceed (tried it on wolfram advanced calculus app and says it doesn't converge)

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There are two different marginal distributions that you can compute. Does Wolfram claim that both integrals are diverge? If only one diverges, which one? –  Dilip Sarwate Apr 16 at 22:06

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The marginal in $x$ is $$ \int_{x^2}^\infty 2xe^{-y} dy = 2x \left. (- e^{-y}) \right|_{x^2}^\infty = 2xe^{-x^2}, \quad x > 0. $$ The marginal in $y$ is $$ \int_0^{\sqrt{y}} 2xe^{-y} dx = e^{-y} \left. (x^2) \right|^{\sqrt{y}}_0 = ye^{-y}, \quad y > 0. $$

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