Area Under a Real Gaussian

Corollary: Setting $p=1/(2\sigma^2)$ in the previous theorem, where $\sigma>0$ is real, we have

$$\int_{-\infty}^\infty e^{-t^2/2\sigma^2}dt = \sqrt{2\pi\sigma^2}, \quad \sigma>0$$ (D.9)

Therefore, we may normalize the Gaussian to unit area by defining

$$f(t) \isdef \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{t^2}{2\sigma^2}}.$$ (D.10)

Since

$$f(t)>0\;\forall t \quad \int_{-\infty}^\infty f(t)\,dt = 1,$$ (D.11)

it satisfies the requirements of a probability density function.