Higher Order Moments Revisited
Theorem:
The
th central moment of the Gaussian pdf
with mean
and variance
is given by
where






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

![\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



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Moment Theorem
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Gaussian Variance