Solutions - 5200-04

By independence, the joint PDF is the product of the marginal PDFs:

$$\begin{array}{c}{f}_X(x)=1,0<x<1f_Y(y)=\frac{1}{2},0<y<2\\ \\ f_{X,Y}(x,y)=\frac{1}{2},0<x<1,0<y<2\end{array}$$