Range of a marginal density function? Given the joint density function: $$p(x,y) = \frac{4x^3}{y^3} \text{ when } 0 < x < 1\text{ and } y > x; \text{ otherwise } p(x,y) = 0$$
How does one calculate the marginal density for $Y$? I do this by simply integration over the the interval in which $X$ lies (so from 0 to 1), which gives $y^{-3}$. But according to my textbook, the answer is $y^{-3}$ for $y > 1$, and $y$ for $0 < y \le 1$? How does that show?

When you integrate with respect to $x$, you must also take into account that $x$ cannot be allowed to exceed $y$.  Sketch the region of integration and you will see what I mean.  Therefore, the correct calculation is $$f_Y(y) = \int_{x=0}^{\min(1,y)} \frac{4x^3}{y^3} \, dx = \frac{\min(1,y^4)}{y^3} = \min(y^{-3},y) = \begin{cases} y^{-3}, & 0 < y \le 1 \\ y, & y > 1. \end{cases}$$
