In the context of harmonic spherical decomposition where $C_\ell$ is the variance of $a_{lm}$ for a given multipole $\ell$, i.e $C_\ell = \langle |a_{lm}^2|\rangle$, I need help about the $\chi^2$ properties.
Actually, I am posting initially this question since I want to compute the mean of obervable $O$, assuming the $a_{lm}$ follows a Normal distribution $\mathcal{N}(0,C_\ell$) since $C_\ell$ is the variance of $a_{lm}$ for a given $\ell$ :
$$O=\frac{\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}}{\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell}a_{\ell m}^{'2}}$$
UPDATE : Maybe there is a solution for the expectation calculation of ratio. Considering the upper and lower terms independent, we could write : E[A/B] = E[A] E[1/B], such that :
$$\begin{aligned} \langle O\rangle &=\left\langle\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell}\left(a_{\ell m}\right)^{2}\right\rangle\left\langle\left(\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell}\left(a_{\ell m}^{\prime}\right)^{2}\right)^{-1}\right\rangle \\ &=\left\langle\sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C_{\ell}\right)\right\rangle\left\langle\left(\sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C_{\ell}^{\prime}\right)\right)^{-1}\right\rangle \end{aligned}$$
1) Given this, by taking the relation between $C_\ell$ and $C'_\ell$ which is : $C_\ell=\bigg(\dfrac{b}{b'}\bigg)^{2}\,C'_\ell$
where $b$ and $b'$ are constants. Warning, rigorously, I should compute for expectation : E[A/B] = E[A] E[1/B]
$$\langle O\rangle = \left(\sum_{\ell=1}^{N} (2\ell+1)\,C_{\ell}\right)\,\langle\left(\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell}\left(a'_{\ell m}\right)^{2}\right)^{-1}\rangle$$
So the expectation E[1/B] is pretty difficult to compute :
If I take : $C_{\ell}=\bigg(\dfrac{b}{b'}\bigg)^{2}\,C'_{\ell}$, then :
$$\langle O\rangle \simeq \dfrac{\sum_{\ell=1}^{N} (2\ell+1)\,\bigg(\dfrac{b}{b'}\bigg)^{2}\,C'_{\ell}}{\sum_{\ell=1}^{N} (2\ell+1)\,C'_{\ell}}=\bigg(\dfrac{b}{b'}\bigg)^{2}$$
Do you think approximation $\simeq$ is not too large or the expression below with Gamma function is more exact ?
2) If I express directly the observable $O$ using $\Gamma$ function, I can get maybe an exact calcultion of $\langle O\rangle$ (I am not totally sure) :
$$\langle O\rangle =\dfrac{\sum_{\ell=1}^{N} \bigg(\dfrac{b}{b'}\bigg)^{2} C_{\ell}^{\prime} \,\Gamma\left((2 \ell+1)/2,2\right)}{\sum_{\ell=1}^{N} C_{\ell}^{\prime} \,\Gamma\left((2 \ell+1)/2,2\right)}=\bigg(\dfrac{b}{b'}\bigg)^{2}$$
So, the mean of observable $O$ would be equal to a constant=$\bigg(\dfrac{b}{b'}\bigg)^{2}$ : is this calculation correct ?
3) To come back on the expression $\left\langle\left(\sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C_{\ell}^{\prime}\right)\right)^{-1}\right\rangle$, the issue is that I can't write :
$$\sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C'_{\ell}\right)=\Gamma\left(\sum_{\ell=1}^{N}(2 \ell+1) / 2,2 C'_{\ell}\right)$$
which could have made things simplier.
Is there a way to simplify this expression $\sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C'_{\ell}\right)$ without removing the dependence in $\ell$ between the shape $(\dfrac{(2\ell+1)}{2}$ ans the scale ($2\,C'_\ell$) parameters ?
