## 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

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.

**Next Section:**

A Sum of Gaussian Random Variables is a Gaussian Random Variable

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Maximum Entropy Property of the Gaussian Distribution