# Poisson noise for a squared contrast of density - expression of variance equal to $1/n^2$

1. I manage to compute the Poisson noise of a density field like this :

If I take $$N$$ the density of galaxies and compute the Shot noise with a Poisson distribution, I get :

$$\langle N^2\rangle - \langle N\rangle^2 = \langle N\rangle$$ so :

$$\langle N^2\rangle = \langle N\rangle + \langle N\rangle^2$$

Let's take the variable : $$X=\dfrac{N}{\langle N\rangle}-1$$

So, I get $$\langle X\rangle=0$$.

Then : $$\langle X^2\rangle = \left\langle\left (\dfrac{N}{\langle N\rangle}-1\right )^2\right\rangle=\dfrac{\langle N^2\rangle}{\langle N\rangle^2}-2+1 = \dfrac{\langle N^2\rangle}{\langle N\rangle^{2}}-1$$

Finally, I get : $$\sigma_x^2 = \langle X^2\rangle=\dfrac{1}{\langle N\rangle}+1-1$$

$$\Rightarrow\quad \sigma_x^2=\dfrac{1}{\langle N\rangle}$$

1. Now, I would like to do the same kind of calculation with variable $$Y=\left(\left(\dfrac{N}{\langle N\rangle}\right)-1\right)^2$$, and conclude normally that :

$$\sigma_y^2 = \dfrac{1}{\langle N\rangle^2}\quad(1)$$

But following the same reasoning with $$N$$ following the Poisson distribution, I can't manage to get this expression $$(1)$$.

UPDATE : i think that I have the proof :

$$\langle X^4\rangle = \Bigg\langle\bigg(\dfrac{N}{\langle N\rangle}-1\bigg)^2\,\bigg(\dfrac{N}{\langle N\rangle}-1\bigg)^2\Bigg\rangle$$

$$=\dfrac{\langle N^2\rangle^2}{\langle N\rangle^4}-2\dfrac{\langle N^2\rangle}{\langle N\rangle^2}+1$$

I) with $$\langle N^2\rangle = \langle N\rangle + \langle N\rangle^2$$

II) and squared :

$$\langle N^2\rangle^2 = \langle N\rangle^2 + 2\langle N\rangle^3 + \langle N\rangle^4$$

We conclude :

$$\langle X^4\rangle=\dfrac{1}{\langle N\rangle^2}+\dfrac{2}{\langle N\rangle} +1 - \dfrac{2}{\langle N\rangle} -2 + 1 = \dfrac{1}{\langle N\rangle^2}$$

• It's not clear to me what you mean by "Shot Noise", particularly as this is usually a synonym for noise that follows Poisson statistics. (e.g., en.wikipedia.org/wiki/Shot_noise) Jun 1 at 12:51
• @uhoh A Poisson distribution has (using the symbols from the Wiki page) an expectation value of $\lambda$ (not $k$) and a variance of $\lambda$. For the "density of galaxies" problem, $\lambda = \bar{n}_{\rm gal}$, so I don't understand why the OP says the "Shot Noise" should be $1/\bar{n}_{\rm gal}$. Jun 2 at 9:26
• @uhoh . Thanks for your answer. Unfortunately, I don't know yet how to justify that the expectation of Shot Noise $\bar{N}_{sp}$ should be equal to 0 like a white noise. I am also interested in the variance of Shot Noise : if I take a fixed value for the spectroscopic Shot Noise, then, the variance is equal to : $<(N_{sp}-\bar{N}_{sp})^{2}>=1/\bar{n}_{\rm gal}^{2}$. Could anyone confirm this result ? Regards Jun 2 at 10:34
• @youpilat13 I'm afraid I have no idea what you're trying to get at. Your last comment mentions "spectroscopic Shot Noise" -- whatever that's supposed to mean -- and then tries to relate that to the mean density of galaxies. That makes no sense. Jun 3 at 11:39
• @youpilat13 Have you tried asking about this on statistics.stackexchange? That seems like a much more appropriate place, since your question appears to be about estimating variances by calculations and manipulation of expectation values, and not really anything astronomical. Aug 11 at 13:03