Suppose I have a random variable $Y$ such that $Y \in \left[ {1 + 1j,1 - 1j, - 1 + 1j, - 1 - 1j} \right]$.
Now by looking at $Y$, I can easily say that $Y$ has a $\pi/2$ circular symmetric distribution. As,
$E[Y] = E[Y{e^{j\frac{\pi }{2}}}]$; $E[{Y^2}] = E\left[ {{{\left\{ {Y{e^{j\frac{\pi }{2}}}} \right\}}^2}} \right]$
Where, $E[.]$ denotes the expectation operation, $j = \sqrt { - 1} $.
Definition: A random variable $Y$ is call $\pi/2$ symmetric if $Y{e^{j\frac{\pi }{2}}}$ has the same distribution (probability distributed function p.d.f.) as $Y$.
My question: My concern is what steps I need to show to prove the $pi/2$ circular symmetry analytically for the following cases:
Known distribution and for a given p.d.f .
Unknown distribution and unknown p.d.f. but conditional p.d.f. is given such as: $p(y|x)$; Where $X \in C$ is say bounded.
I know case 2 is not straightforward as i need to know the distribution of $X$. So, some distribution of $x$ might make $p(y)$ circular symmetrical.