Suppose we have independent normally distributed random variables $\displaystyle X_1 \sim N(c_1,\sigma), X_2 \sim N(c_2,\sigma), \ldots, X_n \sim N(c_n, \sigma)$ where $\displaystyle \sigma$ and all the $\displaystyle c_i$ are known.

Define $\displaystyle A=a_1X_1+\ldots+a_nX_n$ and $\displaystyle B=b_1X_1+\ldots+b_nX_n$ (where all the $\displaystyle a_i$ and $\displaystyle b_i$ are known). Form the complex number $\displaystyle Z=A+Bi$. What would be a good way of computing the distribution of $\displaystyle \arg(Z)$?

I'm not sure how easy it is to find the distribution exactly, so one practical way of trying to go about the problem would be to use simulation. This is fine, but I really want to code up an algorithm that will run quickly - and simulation would be way too slow for what I need. So really I think I need to analytically find the distribution of

$\displaystyle W = \begin{cases} \arctan(B/A) &\mbox{if } X \geq 0 \\ \pi + \arctan(B/A) & \mbox{if } X<0, \end{cases}$

but this doesn't seem particularly easy!