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Let $$Z_k = A\, e^{i(2\pi B+\phi_k)}, \qquad k =1,2,3\dots$$ be a complex function with dependent on $\phi_k$ and others are real constants. Assume that the mean $\mathbb{E}[Z_k] = \mu_{z_k}$ and variance $\mathbb{V}[Z_k] = \sigma_{z_k}^2$ are known.

How I can find the $\mathbb{E}[\phi]$ and $\mathbb{V}[\phi]$ without knowing its distribution? Are there any methods to follow to get these answers?

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With some simple algebra, you obtain the inverse relationship:

$$\phi_k = -i \ln Z_k + i \ln A - 2 \pi B.$$

Hence, taking $A$ and $B$ to be constants (which should really be denoted as lower-case), you have:

$$\begin{equation} \begin{aligned} \mathbb{E}(\phi_k) &= -i \mathbb{E}(\ln Z_k) + i \ln A - 2 \pi B, \\[10pt] \mathbb{V}(\phi_k) &= \mathbb{V}(\ln Z_k). \\[6pt] \end{aligned} \end{equation}$$

So, as you can see, in order to find the desired mean and variance, you need to know the mean and variance of the logarithm of $Z_k$. You have not specified sufficient information in your question to obtain these moment values, so there is no method to get the answer.

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  • $\begingroup$ however in order to find these moments ($\mathbb{E}(\ln Z_k)$ and $\mathbb{V}(\ln Z_k)$), I need the PDF of $Z_k$, is it? but I don't know the PDF of $Z_k$. So then, how do I find them? $\endgroup$ – Ganth Jun 17 '19 at 1:26
  • $\begingroup$ With the information you have given, it is not possible. $\endgroup$ – Ben Jun 17 '19 at 1:33

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