I have a bunch of points in n-d space who's coordinates follow a Normal distribution $(X=x_1,x_2,...,x_n\sim N(0,1) )$.
The coordinates form an angle $\theta$ (with respect to some arbitrary vector $V$) and a magnitude, which I plot.
I calculate the angle of each point using
$$\theta = \cos^{-1}\left( \frac{X \cdot V}{\lvert X \rvert \lvert V \rvert} \right) $$
Then I plot (in 2D) the angles and magnitude for each point.
I am trying to model this distribution in 2D. This way I can sample 2 random variables ($a$ and $b$) and I can calculate the angle with $\tan^{-1}(a/b)$ and the magnitude with $\sqrt{a^2 + b^2}$
At first I figured I would be able to use a bi variate gaussian, but since the magnitude follows a chi-distribution ($\chi_N$), I don't get exactly the same distribution. Since the chi PDF is very low towards zero (for $N > 1$), I get a donut-like distribution in simulation, and a circular distribution with my model.
Is there a better distribution I could use to describe the magnitude? (I'm guessing $a$ and $b$ would be bi-modal gaussians, but how do I go about doing that ?)
PS: I calculate the variances of the components the bivariate gaussians ($a,b$) like so:
\begin{align} \text{var}[\lvert X\rvert^2] &= \text{var}[\chi^2_N] \\ &= \text{var}[a^2 + b^2] \\ &= \text{var}[N(0, \sigma)^2 + N(0, \sigma)^2 ]\\ &= \text{var}[\sigma^2 \ N(0,1)^2 + \sigma^2 \ N(0,1)^2 ]\\ &= \sigma^4 \ \text{var}[N(0,1)^2 + \ N(0,1)^2] \\ &= \sigma^4 \ \text{var}[\chi^2_2]\\ \sigma &= \left( \frac{\text{var}[\chi^2_2]}{\text{var}[\chi^2_N]} \right)^{1/4} \\ \Sigma &= \begin{bmatrix} \sigma^2 & 0\\ 0 & \sigma^2 \end{bmatrix} \end{align}