### Higher Order Moments Revisited

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

 (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

 (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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