# finding marginal distribution - how to determine the limits of integration

This exercise comes from Rice 3.12:

let $$f_{XY}(x,y)=c(x^2-y^2)e^{-x}, 0\leq x <\infty, -x \leq y \leq x$$

b) find the marginal densities

I have found thatc $$c=\frac18$$ and also that $$f_X(x)=\frac16 x^3e^{-x}, 0 \leq x < \infty$$.

for the marginal $$f_Y(y)$$ i wasnt sure how to determine the limits of integration. I have found an solution that states:

for a specific value of $$y$$, $$x$$ ranges from $$|y|$$ to $$\infty$$.

$$f_Y(y)= \int_{|y|}^\infty cx^2e^{-x}dx - \int_{|y|}^\infty cy^2e^{-x}dx$$

Why are there absolute values of y? How are the limits of integration determined?

$$-x \leq y \leq x$$ is equivalent to $$|y| \leq x$$ or $$x \geq |y|$$. when you integrate w.r.t. $$x$$ you have to all the given constraints into account. The only constraint on $$x$$ in this case is $$-x \leq y \leq x$$ so you integrate from $$|y|$$ to $$\infty$$.