# Mean and variance of a variable (inside a function) without known its distribution, but known mean & variance of the function

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?

## 1 Answer

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{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}

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.

• 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? – Ganth Jun 17 '19 at 1:26
• With the information you have given, it is not possible. – Ben Jun 17 '19 at 1:33