TITLE: Integral of the square of the normal distribution.
QUESTION [6 upvotes]: Does someone know what the integral of the square of the normal distribution with standard deviation sigma over the range $-\infty$ to $+\infty$ is? 
Thanks!

REPLY [8 votes]: If you are squaring the PDF...
\begin{align*}
 & \phantom{=}\int_{-\infty}^{\infty}\left(\frac{1}{\sigma\sqrt{2\pi}}\text{exp}\left(-\frac{x^{2}}{2\sigma^{2}}\right)\right)^{2}dx\\
 & =\frac{1}{2\sigma^{2}\pi}\int_{-\infty}^{\infty}\text{exp}\left(-\frac{x^{2}}{\sigma^{2}}\right)dx\\
 & =\frac{\sigma\sqrt{\pi}}{2\sigma^{2}\pi}\\
 & =\frac{1}{2\sigma\sqrt{\pi}}
\end{align*}