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I have a procedure for generating a random complex number $z=|z|e^{i\theta}$ (the procedure is quite complicated, but irrelevant for the question). I want to check whether the modulus and the argument are correlated or not.

I have generated $10^5$ samples. I then ask matlab for the Pearson linear correlation coefficient of $|z|$ and $\theta$ and it gives me $0.0008$. This is indeed quite small, so I conclude for no correlation.

However when I ask for the correlation coefficient of $|z|$ and $e^{i\theta}$, I get $0.15$, which is not really that small.

I am not very familiar with statistics, so maybe my intuition is off. But shouldn't the correlation coefficient be small in both cases? Can $|z|$ be correlated with $e^{i\theta}$ without being correlated with $\theta$?

EDIT

I have looked more closely at the histogram of the joint distribution of $|z|$ and $\theta$ and concluded that they indeed are NOT independent, even though they are uncorrelated. So I know understand what is going on.

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  • $\begingroup$ What sort of joint probability distribution of $|z|$ and $\theta$ was used? $\endgroup$ – Michael Hardy Nov 9 '18 at 17:32
  • $\begingroup$ That is what I want to know. When I make a joint histogram of $z$ and $\theta$, the joint distribution looks like it is a product, so they should be independent. This agrees with correlation being zero between them, but the correlation is not zero between $z$ and $e^{i\theta}$. $\endgroup$ – thedude Nov 9 '18 at 17:45
  • $\begingroup$ Could you please explain how you get a single real value for the correlation coefficient of a real variable and a complex variable? $\endgroup$ – whuber Nov 9 '18 at 18:12
  • $\begingroup$ @whuber the coefficient is actually complex, I just wrote the real part because the imaginary part is much smaller (100 times, roughly) $\endgroup$ – thedude Nov 9 '18 at 18:21
  • $\begingroup$ This raises an interesting question: exactly how do you define $\theta$? Because $\theta$ is really only determined modulo $2\pi,$ it's hard to see how a correlation coefficient between $|z|$ and $\theta$ could have any definite meaning. $\endgroup$ – whuber Nov 9 '18 at 20:53
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This is why in general zero correlation does not imply independence. If $X$ and $Y$ are independent, $E[g(X)h(Y)]=E[g(X)]E[h(Y)]$ in all situations. But, you cannot tell it by only having zero covariance/correlation, i.e. $E[XY]=E[X]E[Y]$.

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