When dealing with real-valued RVs, the extensions of expectation and variance are quite clear to me. For example, showing
$E(aX+b)=aE(X)+b$ and $var(aX+b)=a^2var(X)$ is relatively straightforward to me. Given my weak background in complex numbers, the analogous situation in $\mathbb{C}$ is much more elusive to me.
So, how does one prove (rigorously) the two situations below:
$Z:=X+iY$
$\forall c_1,c_2\in\mathbb{C}, E(c_1Z+c_2)=c_1E(Z)+c_2$
$var(Z)=E(|Z|^2)-|E(Z)|^2=var(X)+var(Y)$
What is especially confusing (and simultaneously fascinating) is that this result apparently is unaffected by the correlation between $X$ and $Y$. How is this possible?