compute marginal I have tried to solve this exercise
Let $X$ and $Y$ be random variables with joint probability density function
given by: $f(x,y)=\frac{1}{8}(x^2-y^2)e^{-x}$ if $x>0$, $|y|<x$
Calculate $E(X\mid Y=1)$
so, the marginal $f_Y(y)$ is $\int_y^\infty \frac{1}{8}(x^2-y^2)e^{-x} dx +\int_{-y}^\infty \frac{1}{8}(x^2-y^2)e^{-x} dx\ $ ?
Is correct?

$$\begin{align}f(x,y)=&~\frac{1}{8}(x^2-y^2)\,\mathsf e^{-x}~\mathbf 1_{x>0, \lvert y\rvert<x}\\[2ex]f_{X\mid Y=1}(x) =&~ \dfrac{f(x,1)~\mathbf 1_{x>1}}{\int\limits_{1}^{\infty}f(s,1)\operatorname d s}\\[1ex] =&~ \dfrac{(x^2-1)\,\mathsf e^{-x}~\mathbf 1_{x>1}}{\int\limits_{1}^{\infty}(s^2-1)\,\mathsf e^{-s}\operatorname d s}\\[1ex] =&~ \tfrac{\mathsf e} 4(x^2-1)\mathsf e^{-x}\mathbf 1_{x>1}\\[3ex] \mathsf E(X\mid Y=1)~=&~\tfrac{\mathsf e} 4\int_1^\infty x(x^2-1)\mathsf e^{-x}\operatorname d x \\=&~\tfrac 7 2\end{align}$$
