I'm dealing with this problem
Suppose that $X, Y$ are iid $N(0,1)$ random variables and let $R= \sqrt{X^2+Y^2}$. Find the joint pdf of $R$ and $R^2$.
I'm not sure how to solve the joint pdf of dependent random variables. I just tried to find cumulative distribution and differentiate it.
By change of variables, I got $f_R(r) = \frac{1}{2\pi}r\exp{-\frac12r}$. $$f_{R,R^2}(x,y) = \frac{\partial^2}{\partial x \partial y} P(R \le x, R^2 \le y)= \frac{\partial^2}{\partial x \partial y} P(R \le x , -\sqrt{y} \le R \le \sqrt{y} ) $$ so this is nonzero only if $-\sqrt{y} \le x \le \sqrt{y}$, and this is equal to $\frac{\partial^2}{\partial x \partial y} \int^x_{-\sqrt{y}} f_R(r) dr =0$, but this doesn't make sense. How should I deal with this?