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Area Under a Real Gaussian
Corollary:
Setting
in the previous theorem, where
is real,
we have
![$\displaystyle \int_{-\infty}^\infty e^{-t^2/2\sigma^2}dt = \sqrt{2\pi\sigma^2}, \quad \sigma>0$](img2743.png) |
(D.9) |
Therefore, we may normalize the Gaussian to unit area by defining
![$\displaystyle f(t) \isdef \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{t^2}{2\sigma^2}}.$](img2744.png) |
(D.10) |
Since
and![$\displaystyle \quad \int_{-\infty}^\infty f(t)\,dt = 1,$](img2746.png) |
(D.11) |
it satisfies the requirements of a probability density function.
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