How can I analytically derive $E[\alpha_1\alpha_2]$, where $X_1=X_{1r}+j\,X_{1i}$ and $X_2=X_{2r}+j\,X_{2i}$. Further, $\alpha_1=|X_1|$ and $\alpha_2=|X_2|$?
Here, $X_{1r,},X_{1i},X_{2r,},X_{2i}\sim N(0,\sigma^2)$.
Also $X_{1r,}$ and $X_{1i}$ are independent, $X_{2r}$ and $X_{2i}$ are independent, and $X_1$ and $X_2$ have correlation coefficient $\rho$.
I wanted this when I tried to derive $Var(\alpha_1\alpha_2)$. We can assume angles of $X_1$ and $X_2$ are uniform in $[0,2\pi)$.