Gaussian Variance
The
variance of a distribution
is defined as its
second central moment:
![]() |
(D.43) |
where
![$ \mu$](http://www.dsprelated.com/josimages_new/sasp2/img385.png)
![$ f(t)$](http://www.dsprelated.com/josimages_new/sasp2/img560.png)
To show that the variance of the Gaussian distribution is
, we write,
letting
,
![\begin{eqnarray*}
\int_{-\infty}^\infty (t-\mu)^2 f(t) dt &\isdef &
g \int_{-\infty}^\infty (t-\mu)^2 e^{-\frac{(t-\mu)^2}{2\sigma^2}} dt\\
&=&g \int_{-\infty}^\infty \nu^2 e^{-\frac{\nu^2}{2\sigma^2}} d\nu\\
&=&g \int_{-\infty}^\infty \underbrace{\nu}_{u} \cdot \underbrace{\nu e^{-\frac{\nu^2}{2\sigma^2}} d\nu}_{dv}\\
&=& \left. g \nu (-\sigma^2)e^{-\frac{\nu^2}{2\sigma^2}} \right\vert _{-\infty}^{\infty} \\
& & - g \int_{-\infty}^\infty (-\sigma^2) e^{-\frac{\nu^2}{2\sigma^2}} d\nu \\
&=&\sigma^2
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2831.png)
where we used integration by parts and the fact that
as
.
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Higher Order Moments Revisited
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Gaussian Mean