$Z$ is a random variable that follows a standard normal distribution. Is $X = Z^2$ independent from $Y = Z^3$? I think the title is pretty clear about the problem. Should I try to find the joint probability of $X$ and $Y$ and decide if $\, f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$? If so, how am I going to find a joint distribution like this? Or is there some shortcut to prove it?

$EXY^{2} \neq EXEY^{2}$ so $X$ and $Y^{2}$ are not independent. This implies that $X$ and $Y$ are not independent. [Use a table of moments of $Z$ to see that $EXY^{2} \neq EXEY^{2}$. You can also argue analytically using that fact that 6-th moment is strictly smaller than the 8th moment].  Alternative proof: if $X$ and $Y$ are independent so are $X^{3}$ and $Y^{2}$. This makes $Z^{6}$ independent of itself which implies $Z^{6}$ is a constant random variable. This is obviously false. 
