Show that for a Gaussian probability distribution [p(x)=frac{e^{-frac{left(x-x_{0} ight)^{2}}{2 a^{2}}}}{sqrt{2 pi a^{2}}}] all the moments are given

Show that for a Gaussian probability distribution

\[p(x)=\frac{e^{-\frac{\left(x-x_{0}\right)^{2}}{2 a^{2}}}}{\sqrt{2 \pi a^{2}}}\]

all the moments are given by \[\left\langle\left(x-x_{0}\right)^{n}\rightangle=1 \times 3 \times 5 \times(n-1) \times a^{n}\]
for even \(n\), and are zero otherwise. Hence the Gaussian distribution is entirely characterized by its mean \(x_{0}\) and deviation \(a\).