### 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]:

(D.46) |

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

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