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I am trying to prove that a complex variable $Z = R.\exp(i.\Phi) = R.\cos(\Phi) + i.R.\sin(\Phi) = X + i.Y$ does not follow a normal distribution when $R\sim \mathcal{N}\left(\mu_R, \sigma_R^{2}\right)$ and $\Phi \sim \mathcal{N}\left(\mu_\Phi, \sigma_\Phi^{2}\right)$.

Any ideas on how to go about this? I tried computing the expectation value E[z] but that seems to be a dead end. Thank you in advance.

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Assuming $R,\,\Phi$ are independent, their joint pdf $f_{R,\,\Phi}(r,\,\phi)$ satisfies $f_{R,\,\Phi}(r,\,\phi)drd\phi=f_{X,\,Y}(x,\,y)dxdy$, with $f_{X,\,Y}$ the joint pdf of $X,\,Y$. Since $dxdy=rdrd\phi$, $$f_{X,\,Y}(x,\,y)=r^{-1}f_{R,\,\Phi}(r,\,\phi)=\frac{1}{2\pi r\sigma_R\sigma_\Phi}\exp-\left[\frac{(r-\mu_R)^2}{2\sigma_R^2}+\frac{(\phi-\mu_\Phi)^2}{2\sigma_\Phi^2}\right].$$ I'm sure you can show, be rewriting this in terms of $x,\,y$, it's not even close to being bivariate normal.

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