Let $\displaystyle Z = X + iY$ be complex random variable. Prove that:
$\displaystyle |EZ| \le E|Z|$
First of all we do not know if joint density for the variables exists. However I know (somebody told me) that it can be done in more abstract way using:
1. $\displaystyle X \le Y \Rightarrow EX \le EY$ (for real variables)
2. Cauchy-Schwartz inequality
But I don't know how....
Solution
Let $\displaystyle \phi = \arg EZ$
Than: $\displaystyle |EZ| = EZe^{-i\phi} = E(X+iY)(\cos\phi - i \sin \phi) = $
$\displaystyle =E(X\cos\phi + Y \sin \phi) + i E(Y\cos\phi - X \sin \phi)$
Because $\displaystyle |EZ| \in \mathbb{R}$ we have $\displaystyle E(Y\cos\phi - X \sin \phi) = 0$
Hence by Cauchy-Schwartz inequality (and expected value monotonicity)
$\displaystyle
|EZ| = E(X\cos\phi + Y \sin \phi) \le E\left(\sqrt{X^2+Y^2}\sqrt{\cos^2\phi+\sin^2\phi}\ right) = E|Z|$