here I have a $M\sim \mathbb{N}(0,\sigma^2_N)$ and $$ M = \int \int_A \frac{\partial}{\partial x} \Bigl(\int^\infty f(x,z,\ell) d \ell\Bigl) dx dz $$ where $f(x,z,\ell)$ is also a r.v. I would like to obtain second moment of $M$ which is ($\mathbb{E}[M^2]$).
I know that I should either apply Leibniz integral rule on $M$, or use the charecteristic function to get the second moment directly which goes beyond my calculus knowledge. Of course, the derivation steps to the closed expression is all I need. I also have the supposed result:
$$ \mathbb{E}[M^2] = \int \int_A dx dz \int \int_A dx' dz' \int_0^\infty d\ell \int_0^\infty d \ell' \frac{\partial^2}{\partial x \partial x'} \mathbb{E}[f(x,z,\ell) f(x',z',\ell')] $$
I would appreciate if you would direct and help me.
bowellarge