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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Gaussian Function Properties
       Gaussian Moments
          Gaussian Variance

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Gaussian Variance

The variance of a distribution $ f(t)$ is defined as its second central moment:

$\displaystyle \sigma^2 \isdef \int_{-\infty}^\infty (t-\mu)^2 f(t)dt $

where $ \mu$ is the mean of $ f(t)$.

To show that the variance of the Gaussian distribution is $ \sigma^2$, we write, letting $ g\isdef 1/\sqrt{2\pi\sigma^2}$,

\begin{eqnarray*} \int_{-\infty}^\infty (t-\mu)^2 f(t) dt &\isdef & g \int_{-\in... ...ty (-\sigma^2) e^{-\frac{\nu^2}{2\sigma^2}} d\nu \\ &=&\sigma^2 \end{eqnarray*}

where we used integration by parts and the fact that $ \nu f(\nu)\to 0$ as $ \left\vert\nu\right\vert\to\infty$.

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Next: Higher Order Moments Revisited

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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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