$X_1,X_2$ iid standard normal with polar coordinates r and p. Are r and p independent?

Question

I have two scalar random variables $X_1$ and $X_2$ and they are both independent and both have standard normal distribution $N(0,1)$. I am letting $r$ and $p$ be the polar coordinates of the point $(X_1,X_2)$, so this means that $X_1=rcos(p)$ and $X_2=rsin(p)$. I am trying to decide if the random variables $r$ and $p$ are independent. I think that they are not but I could be wrong, anyone see why they are or are not?

Answer 1

Since $X_1$ and $X_2$ are independent, their joint pdf is $$ f(x_1,x_2)=\frac{1}{2\pi}e^{-\frac{x_1^2+x_2^2}{2}} $$

Next, we have $R=\sqrt{X_1^2+X_2^2}$ and $\Theta=\arctan(\frac{Y}{X})$. One can show that the Jacobian of this transformation is $\frac{1}{R}$, hence the joint pdf of $R$ and $\Theta$ is $$ f(r,\theta)=\frac{r}{2\pi}e^{-\frac{r^2}{2}} $$ for $r\geq0$ and $0\leq \theta\leq 2\pi$. This is the product of the pdfs of $R$ and $\Theta$, so $R$ and $\Theta$ are independent.