(Updated) How to compute expectation and variance of an argument of a complex random variable? Assume that $\xi$ is a complex random variable. Its argument $\arg \xi$ is a real random variable. I am interested in how to computed expectation and variation of $\arg \xi$.
Edit: 
I add more specifics to the question. Let $\varepsilon_j$ is a normal random variable with zero mean and $\sigma^2_j$ variance. I am interested in case when $$ \xi = \sum_j a_j\exp(i\varepsilon_j)$$ where $a_j$ are some real value constants. How to compute variance and expectation of $\arg \xi$?.
 A: This seems to be a simple application of the law of the unconscious statistician.  That is, suppose $\Xi$ is a random variable with pdf $f(\xi)$.  Then we have
$$
E(\arg(\Xi)) = \iint_\mathbb{C} \arg(\xi)f(\xi)\,d\xi
$$
In order to put that in terms that you might be more familiar with: let $X = \operatorname{Re}\{\Xi\}$ and $Y = \operatorname{Im}\{\Xi\}$ so that $\Xi = X + iY$.  Similarly, let $x$ and $y$ be real variables and define $\xi = x+iy$.  Then, define $f_R(x,y)=f(x+iy)$.  Note that $\arg(x+iy)$ may be computed as described here.  From there, we may calculate the expectation as a two-dimensional real integral.  Namely,
$$
E(\arg(\Xi)) =
E(X + iY) = 
\iint_{\mathbb{R}^2} \arg(x+iy)f_{R}(x,y)\,dx\,dy\\
= \int_{-\infty}^\infty\int_{-\infty}^\infty
\arg(x+iy)f_{R}(x,y)\,dx\,dy
$$
I hope that's the answer you were looking for.

As for variance, the process is similar.  Simply calculate $E(\arg^2(\Xi))-E(\arg(\Xi))^2$, using the above calculation of $E(\arg(\Xi))$ and the same process to calculate $E(\arg^2(\Xi))$.
As for the newly added specifics of your question, I will attempt a solution if I can.
