Ytukyg

Represent $g sim N(0,I_n)$in polar form as $g=r theta$ where $r = |g|_2$ is the length and $theta = frac{g}{|g|_2} $ is the direction

prove that $r$ and $theta$ are independent ?

prove that $theta$ is uniformly distributed on sphere $S^{n-1}$

for first one :
The only things I know how to do is to show the product pdf of both of $theta$ and $r$ is same as n-dimeinal pdf of of standard Gaussian vector ? but how to find pdf of $|g|_2$? I have some problem in trasformation of random variable in this case . since the transformationare not bijective , is any simple way to do that?

Big hint: The joint density of $G = (X, Y)$ is $f(x,y) , dx , dy = frac{1}{2pi} e^{-(x^2+y^2)/2} , dx , dy$. By performing a change of variables to polar coordinates $R^2 = X^2 + Y^2$ and $Theta = arctan(Y/X)$, we have $$f(r, theta) , dr , dtheta = frac{1}{2pi} e^{-r^2/2} r , dr , dtheta,$$
where the extra factor of $r$ comes from the Jacobian when performing the change of variables.

Hint: If you know the joint density can be written as $f(r, theta) = g(r) h(theta)$ for some densities $g$ and $h$, then $R$ and $Theta$ are independent.