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Variance



Definition: The variance or second central moment of a stochastic process $ v(n)$ at time $ n$ is defined as the expected value of $ \left\vert v(n)-\mu_{v(n)}\right\vert^2$ :

$\displaystyle \sigma^2_{v(n)} \isdef E\{\left\vert v(n)-\mu_{v(n)}\right\vert^2\} \isdef \int_{-\infty}^\infty \left\vert v(n)-\mu_{v(n)}\right\vert^2 p_{v(n)}(x) dx$ (C.19)

where $ p_{v(n)}(x)$ is the probability density function for the random variable $ v(n)$ .

For a stationary stochastic process $ v$ , the variance is given by the expected value of $ \left\vert v(n)-\mu_v\right\vert^2$ for any $ n$ .


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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