Let $U,V$ be two independent random variables.
$U$ ~ $Uniform(0,2 \pi) $ and $V$ ~ $Exp(\frac{1}{2})$.
We consider $(X,Y) \in \mathbb{R^2} $ with polar coordinates $R = \sqrt(V)$ and $\phi=U$.
So $X = R \cdot cos(\Phi)$ and $Y = \cdot sin(\Phi)$.
Calculate the joint probability density function for $X,Y$.
My work:
So I know the pdf of $U$ and I know the pdf of $V$.
I can start with $ P( (X,Y) \in A) = P( (\sqrt(V) \cdot cos(\Phi) ,\sqrt(V) \cdot sin(\Phi)) \in A)= P((\sqrt(V),\Phi) \in A')$. So as you can see I will need the joint pdf of $\sqrt(V)$ and $U$. I already compute the pdf of $\sqrt(V)$. But my problem is: are $\sqrt(V)$ and $U$ independent? We know that that $V$ and $U$ are independent.