Your function does not appear to be a PDF -- the integral (over all x, or x, n) is greater than 1. In fact it is very close to
In:= PDF[NormalDistribution[0, \[Pi]^(1/4) \[Sigma]], x]
Out= E^(-(x^2/(2 Sqrt[\[Pi]] \[Sigma]^2)))/(Sqrt \[Pi]^(
which is different by a factor of
I hope that helps indicate where the problem is.