Let X and Y be continuous, with joint pdf: \[ f_{X,Y} = \begin{cases} 2, & \text{if 0 < x < 1, 0 < y < x} \\ 0, & \text{otherwise} \\ \end{cases} \] Let $Z = XY^3$. What is the pdf of Z?
I was thinking that maybe I could let W be another function of X and Y and finding the joint density of Z and W, then solving for the marginal density of Z.