Let $R$ and $U$ be independent random variables where $R$ has density $f_{R}(r)=2 r\cdot1_{[0,1]}(r)$ and $U$ is the uniform distribution on $[0,1]$. Furthermore let $X= R \cos{(2\pi U)}$ and $Y=R \sin{(2\pi U)}$. Now, determine the density function of $(X,Y)$.
I do not know where to begin, and even if I find some function, how can I be sure that it is the density function.
The density of $X$ is $f_{X}(x)=2 x\cdot1_{[0,1]}(x)\cos{(2\pi x)}$ or is it $f_{X}(x, \overline{x})=2x \cdot1_{[0,1]}(x)\cos{(2\pi \overline{x})}$ because of independence of $R$ and $U$? And how exactly can I compute $f_{(X,Y)}$?
Let $f_{X,\,Y}(x,\,y)$ denote the joint PDF so $$f_{X,\,Y}(x,\,y)dxdy=f_R(r)f_\Theta(\theta)drd\theta=\frac{r}{\pi}1_{[0,\,1]}(r)drd\theta.$$Dividing by $dxdy=rdrd\theta$,$$f_{X,\,Y}(x,\,y)=\frac{1}{\pi}1_{[0,\,1]}(r)=\frac{1}{\pi}[x^2+y^2\le1].$$In other words, it's a uniform density on the unit circle.