Gaussian Moments

Gaussian Mean

The mean of a distribution is defined as its first-order moment: (D.42)

To show that the mean of the Gaussian distribution is , we may write, letting , since .

Gaussian Variance

The variance of a distribution is defined as its second central moment: (D.43)

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 .

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 .

Moment Theorem

Theorem: For a random variable , (D.47)

where is the characteristic function of the PDF of : (D.48)

(Note that is the complex conjugate of the Fourier transform of .)

Proof: [201, p. 157] Let denote the th moment of , i.e., (D.49)

Then where the term-by-term integration is valid when all moments are finite.

Gaussian Characteristic Function

Since the Gaussian PDF is (D.50)

and since the Fourier transform of is (D.51)

It follows that the Gaussian characteristic function is (D.52)

Gaussian Central Moments

The characteristic function of a zero-mean Gaussian is (D.53)

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 (D.54)

In particular, Since and , we see , , as expected.

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Maximum Entropy Property of the Gaussian Distribution