To show that the mean of the Gaussian distribution is , we may write, letting ,
where is the mean of .
To show that the variance of the Gaussian distribution is , we write, letting ,
where we used integration by parts and the fact that as .
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 .
Theorem: For a random variable ,
where is the characteristic function of the PDF of :
(Note that is the complex conjugate of the Fourier transform of .)
Proof: [201, p. 157] Let denote the th moment of , i.e.,
where the term-by-term integration is valid when all moments are finite.
Gaussian Characteristic Function
Since the Gaussian PDF is
and since the Fourier transform of is
It follows that the Gaussian characteristic function is
The characteristic function of a zero-mean Gaussian is
Since a zero-mean Gaussian is an even function of , (i.e., ), all odd-order moments are zero. By the moment theorem, the even-order moments are
Since and , we see , , as expected.
A Sum of Gaussian Random Variables is a Gaussian Random Variable
Maximum Entropy Property of the Gaussian Distribution