I began studying probability theory some time ago and now I've reached multiple random variables and function of random variables and I'm stuck on this particular problem. The issue here is, even though I've solved other problems, I've no idea how to approach the following:
Let $(X_1, X_2)$ be a random variable with PDF (probability density function):
$$f(x_1, x_2)=\frac{1}{2\pi}e^{\frac{x_1^2 + x_2^2}{2}} \in \mathbb{R}$$
and let $X1 = Y_1\cos{Y_2}$ and $X_2 = Y_1\sin{Y_2}$, with $Y_1\gt0$. Prove that $Y_1, Y_2$ are independent and calculate their PDFs.
If you consider the function $f(x_1, x_2)=\frac{1}{2\pi}e^{-\frac{x_1^2 + x_2^2}{2}}$ on the plane $(x_1,x_2)$, it depends only on the distance from the center $y_1=\sqrt{x_1^2+x_2^2}$ . It is intuitively clear from here that the marginal PDF of $Y_2$ (polar angle) is uniform on $[0,2\pi]$ and $Y_1$ does not depend on $Y_2$.
The probability that $Y_1$ takes value in a small ring $(y_1,y_1+dy_1)$ where $y_1^2=x_1^2+x_2^2$ is equal to $$f_{Y_1}(y_1)dy_1=f(x_1,x_2)\cdot 2\pi y_1\cdot dy_1$$ so $$f_{Y_1}(y_1)=f(x_1,x_2)\cdot 2\pi y_1=y_1e^{-\frac{y_1^2}{2}}$$