Area Under a Real Gaussian

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

$\displaystyle \int_{-\infty}^\infty e^{-t^2/2\sigma^2}dt = \sqrt{2\pi\sigma^2}, \quad \sigma>0$

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

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

Since

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

it satisfies the requirements of a probability density function.

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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