# Compute Variance of a the ratio of different terms : which term to consider as random variable?

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.

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}
• @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 $X$.