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
successively with respect to [203, p. 147-148]:
for . Setting and , and dividing both sides by yields
for . Since the change of variable has no affect on the result, (D.44) is also derived for .