## 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