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Gaussian Variance

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

$\displaystyle \sigma^2 \isdef \int_{-\infty}^\infty (t-\mu)^2 f(t)dt$ (D.43)

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}$ ,

\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 \\

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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