## 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