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?

The best way to understand this answer is to draw out a graph showing the ranges of $x$ and $y$. Draw a vertical line at $x=1$ to show that boundary value and draw the line $y=x$. Now shade in the region that corresponds to $y>x$. You should be able to see that the range of $y$ can be divided into two parts: $y>1$ and $0<y\le1$. Hopefully the limits of integration for each part should be clear by this point.
