When you have done part a) and determined the distribution for $\displaystyle X$ and $\displaystyle Y$ you can use the fact that $\displaystyle X$ and $\displaystyle Y$ are independent iff.
$\displaystyle F_{X,Y}(x,y) = F_X(x)F_Y(y)$ or $\displaystyle f_{X,Y}(x,y) = f_X(x)f_Y(y)$
Where $\displaystyle f_{X,Y}$ are the joint p.d.f of $\displaystyle X$ and $\displaystyle Y$ and $\displaystyle F_{X,Y}$ are the respective the c.d.f