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Theorem:
The
th central moment of the Gaussian pdf
with mean
and variance
is given by
![$\displaystyle m_n \isdef {\cal E}_p\{(x-\mu)^n\} = \left\{\begin{array}{ll} (n-1)!!\cdot\sigma^n, & \hbox{$n$\ even} \\ [5pt] $0$, & \hbox{$n$\ odd} \\ \end{array} \right. \protect$](img2825.png) |
(D.44) |
where
denotes the product of all odd integers up to and
including
(see ``double-factorial notation'').
Thus, for example,
,
,
, and
.
Proof:
The formula can be derived by successively differentiating the
moment-generating function
with respect to
and evaluating at
,D.4 or by differentiating the
Gaussian integral
![$\displaystyle \int_{-\infty}^\infty e^{-\alpha x^2} dx = \sqrt{\frac{\pi}{\alpha}}$](img2833.png) |
(D.45) |
successively with respect to
[203, p. 147-148]:
for
.
Setting
and
, and dividing both sides by
yields
![$\displaystyle {\cal E}_p\{x^n\} \isdefs \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^\infty x^n e^{-\frac{x^2}{2\sigma^2}} dx \eqsp \zbox {\sigma^n \cdot (n-1)!!}$](img2839.png) |
(D.46) |
for
. Since the change of variable
has no affect on the result,
(D.44) is also derived for
.
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