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I have the following expression with random variable $X$ :

$X = \bigg(\dfrac{b_1^2}{b_2^2} + \dfrac{N(\ell)}{f(\ell)}\bigg)$

with $b_1$ and $b_2$ constants and $\ell$ a variable.

Can I write :

$\sigma_x^2 = \text{Var}\bigg(\dfrac{N(\ell)}{f(\ell)}\bigg)$

?

I know that $\text{Var}(aX) = a^2 \text{Var}(X)$ but I don't know which variable $X$ to use ? ($N(\ell)=N$ is a Poisson noise actually independent from $\ell$)

Normally, I should have :

$Var(X) = \text{Var}\bigg(\dfrac{N}{f(\ell)}\bigg)$

that is to say, by using : $\text{Var}(X) = \text{E}[X^2]-\text{E}[X]^2$ , but I can't conclude with this relation in my case for X since I don't know which term as random variable to use.

UPDATE : I bring other informations, rather physical, but all this is mostly about the topic of Legendre transformation with $C_{\ell}$ and multipole $\ell$.

The covariance from which I am expected to compute the error on Galaxy-Galaxy observable is given below :

Using the ratio of Power spectra :

Below the error on photometric galaxy clustering under the form of covariance : $$ \Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{A B}(\ell)\right]\quad(1) $$ where $f_{\text {sky }}$ is the fraction of surveyed sky and $A, B$ run over the observables $L$ and $G, \Delta \ell$ is the width of the multipoles bins used when computing the angular power spectra, and $i, j$ run over all tomographic bins. The First term $C_{i j}^{A B}$ refers to the Cosmic Variance and the second term $N_{i j}^{A B}(\ell)$ is the Shot Noise (Poisson noise). We look at here $A, B=G$. Taking the ratio between both on (19), one can write: $$ O_{2}=\left(\frac{b_{s p}^{2} C_{\ell, \mathrm{DM}}^{\prime}+\Delta C_{s p, i j}^{G G}}{b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime}+\Delta C_{p h, i j}^{G G}}\right)\quad(2) $$ We neglect the Poisson noise term $\Delta C_{p h, i j}^{G G}$ (sum of Cosmic Variance and Shot Noise) $\Delta C_{p h, i j}^{G G}$ on denominator since it is very small compared to $b_{p h}^{2} C_{\ell, \mathrm{DM}}^{\prime} .$ We consider also the dominance of spectroscopic Shot Noise $N_{s p, i j}^{G G}(\ell)$ in the quantity $\Delta C_{s p, i j}^{G G}$

I think the equation $(2)$ is not valid since I include the error in the observable itself whereas I just want to estimate the error o the ratio :

$$O_{2}=\left(\dfrac{b_{sp}^{2}}{b_{ph}^{2}}\right)\quad(3)$$.

QUESTION :Anyone could have an idea to esimate this error of $O_{2}$ in equation($3$) ?

I have to start from the definition of $C_{\ell}$ to compute angular power spectrum appearing in equation($1$) and equation($2$) : this latter I recall it, I think it's false.

Sorry for the Physics context, on physics echange, I have had no answers and I thought that it was mostly some mathematical stuff with the Legendre transformation and statistics.

Any help is welcome.

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The correct calculation gives a somewhat different result:$$\begin{align}\operatorname{Var}X&=\Bbb E(X^2)-(\Bbb EX)^2\\&=\Bbb E(b_1^2/b_2^2+N/f)-\left(\Bbb E\sqrt{b_1^2/b_2^2+N/f}\right)^2\\&=b_1^2/b_2^2+\Bbb E(N/f)-\left(\Bbb E\sqrt{b_1^2/b_2^2+N/f}\right)^2.\end{align}$$

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  • $\begingroup$ @yourpilat13 Subtracting a constant from $X$ preserves the variance of $X$; subtracting a constant from $X^2$ preserves the variance of $X^2$, but not of $X$. $\endgroup$
    – J.G.
    Apr 3 '21 at 15:27
  • $\begingroup$ I have modified my question, I ask for which term to consider to compute the variance. $\endgroup$
    – youpilat13
    Apr 11 '21 at 8:29

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