Higher Order Moments Revisited
Theorem:
The
th central moment of the Gaussian pdf
with mean
and variance
is given by
where
![$ (n-1)!!$](http://www.dsprelated.com/josimages_new/sasp2/img2835.png)
![$ n-1$](http://www.dsprelated.com/josimages_new/sasp2/img2592.png)
![$ m_2=\sigma^2$](http://www.dsprelated.com/josimages_new/sasp2/img2836.png)
![$ m_4=3\,\sigma^4$](http://www.dsprelated.com/josimages_new/sasp2/img2837.png)
![$ m_6=15\,\sigma^6$](http://www.dsprelated.com/josimages_new/sasp2/img2838.png)
![$ m_8=105\,\sigma^8$](http://www.dsprelated.com/josimages_new/sasp2/img2839.png)
Proof:
The formula can be derived by successively differentiating the
moment-generating function
with respect to
and evaluating at
,D.4 or by differentiating the
Gaussian integral
![]() |
(D.45) |
successively with respect to
![$ \alpha $](http://www.dsprelated.com/josimages_new/sasp2/img4.png)
![\begin{eqnarray*}
\int_{-\infty}^\infty (-x^2) e^{-\alpha x^2} dx &=& \sqrt{\pi}(-1/2)\alpha^{-3/2}\\
\int_{-\infty}^\infty (-x^2)(-x^2) e^{-\alpha x^2} + dx &=& \sqrt{\pi}(-1/2)(-3/2)\alpha^{-5/2}\\
\vdots & & \vdots\\
\int_{-\infty}^\infty x^{2k} e^{-\alpha x^2} dx &=& \sqrt{\pi}\,[(2k-1)!!]\,2^{-k/2}\alpha^{-(k+1)/2}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2843.png)
for
.
Setting
and
, and dividing both sides by
yields
![]() |
(D.46) |
for
![$ n=2,4,6,\ldots\,$](http://www.dsprelated.com/josimages_new/sasp2/img2849.png)
![$ x
= \tilde{x}-\mu$](http://www.dsprelated.com/josimages_new/sasp2/img2850.png)
![$ \mu\ne0$](http://www.dsprelated.com/josimages_new/sasp2/img2851.png)
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Moment Theorem
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Gaussian Variance