### Higher Order Moments Revisited

**Theorem: **
The
th central moment of the Gaussian pdf
with mean
and variance
is given by

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

(D.45) |

successively with respect to [203, p. 147-148]:

for . Setting and , and dividing both sides by yields

(D.46) |

for . Since the change of variable has no affect on the result, (D.44) is also derived for .

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Moment Theorem

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Gaussian Variance