# Integrating a product of Gaussian distributions

I'm stuck at this relatively easy looking integral where I have gaussian distributions $$\begin{equation} \sigma(x,y)=\frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2+y^2}{4\sigma^2}} \end{equation}$$ and the integral $$\begin{equation} I(x)=\int_{-\infty}^{\infty}dx_N \cdots \int_{-\infty}^{\infty}dx_0 \delta(x-\sum_{i=0}^{N}x_i)\sigma(x_0-\mu_0,x_0-\mu'_0)\sigma(x_1-\mu_1,x_1-\mu'_1)\cdots \sigma(x_N-\mu_N,x_N-\mu'_N). \end{equation}$$

I tried putting this on mathematica and I got a distribution $$I(x)$$ that is again a gaussian with the variance now scaled with $$N$$. I did this only for specific $$N$$ on mathematica and inferred the scaling. Would appreciate if anybody could guide me on how to do this integral for any $$N$$.

• Have you heard about Central limit theorem? – user35952 Oct 4 at 11:02
• @user35952: Not very helpful in this case. It's obvious that the integral is going to result in a gaussian. What is not obvious is the prefactors that emerge after completing the squares and integrating. – Juzar Oct 7 at 2:18