I have the following integration problem,
$F = \int_a^b \int_a^b \mathcal{N}(x;y,\sigma_0)\mathcal{N}(x;0,\sigma_1)\mathcal{N}(y;0,\sigma_1) dx dy $,
where $\mathcal{N}(x;y,\sigma_0)$ is a gaussian distibution over $x$ with mean $y$ and std. $\sigma_0$.
I want to find $F$ after integration has been performed. Assume $a,b,\sigma_0,\sigma_1$ are all constants. I have tried a few tactics (replacing normals with their taylor series, solving the first integral and converting the ERF to a taylor series, approximating gaussian function by functions of $tanh$) but I keep getting stuck.
Any help or the solution would be greatly appreciated! If there is no obvious way to solve this a good approximation may suffice.
$N(x,y, \sigma_0) = \frac{e^{- \frac{(x-y)^2}{2 \sigma_0^2}}}{\sqrt{2 \pi \sigma_0^2}} = \frac{e^{- \frac{x^2-2xy+y^2}{2 \sigma_0^2}}}{\sqrt{2 \pi \sigma_0^2}}$
$N(x,0, \sigma_1) N(x,y, \sigma_0) N(y,0, \sigma_2) = \frac{e^{- \frac{x^2-2xy+y^2}{2 \sigma_0^2} - \frac{x^2}{2 \sigma_1^2} - \frac{y^2}{2 \sigma_2^2}}}{\sqrt{(2 \pi)^3 \sigma_0^2 \sigma_1^2 \sigma_2^2}}$.
Now you can put $x,y$ into quadratic Terms like $(x-a)^2,(y-b)^2)$ and evaluate the Gaussian integrals separately.