Why is the standard deviation $\displaystyle\sigma$ defined in such a way that in the exponent of the normal distribution,
$\displaystyle f{{\left({x}\right)}}=\frac{1}{{\sigma\sqrt{{{2}\pi}}}}{e}^{{-{\left(\frac{{{x}-\mu}}{{\sigma\sqrt{{{2}}}}}\right)}^{2}}}$
$\displaystyle \sigma$ needs to be scaled up by an additional factor of $\displaystyle\sqrt{{{2}}}$?
Because intuitively, I would define the normal distribution like this, namely simply as the normalized Gaussian integral:
$\displaystyle {\int_{{-\infty}}^{{+\infty}}}{e}^{{-{x}^{2}}}{\left.{d}{x}\right.}=\sqrt{{\pi}}\quad\Rightarrow\quad\displaystyle\ f{{\left({x}\right)}}:\:=\frac{1}{\sqrt{{\pi}}}{e}^{{-{x}^{2}}}$