Could anybody explain how to prove this? $$\int \delta(x_{1}+x_{2}+....+x_{n}-\eta{\sqrt n} ) \prod p(x_{i})dV = \frac{\exp(-\eta^2/2)}{\sqrt2\pi}$$
When $x_{i}$ (i=1,2,....,n)is independently follow standard normal distribution, dV means volume element and $\delta$ means dirac's delta function.
I thought this could be proved by using the fact that unit n-1 sphere has the surface area of $$S_{n-1}=\frac{2\pi^\frac{n}{2}}{\Gamma(n/2)}$$
but, I couldn't prove this.
I thought the above integral can be written as $$\int\int \delta({\sqrt n}(r-\eta) )S_{n-1}z^{n-1}\frac{e^{\frac{-z^2-r^2}{2}}}{(2\pi)^{n/2}} drdz$$by replacing $\sum_{i=1}^{n}x_i$ with $r \sqrt{n}$. But the answer I got was $e^{\frac{-\eta^2}{2}}/\sqrt{n}$. I would appreciate it if someone could explain.