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


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






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

![\begin{eqnarray*}
\int_{-\infty}^\infty (-x^2) e^{-\alpha x^2} dx &=& \sqrt{\pi}(-1/2)\alpha^{-3/2}\\
\int_{-\infty}^\infty (-x^2)(-x^2) e^{-\alpha x^2} + dx &=& \sqrt{\pi}(-1/2)(-3/2)\alpha^{-5/2}\\
\vdots & & \vdots\\
\int_{-\infty}^\infty x^{2k} e^{-\alpha x^2} dx &=& \sqrt{\pi}\,[(2k-1)!!]\,2^{-k/2}\alpha^{-(k+1)/2}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2843.png)
for
.
Setting
and
, and dividing both sides by
yields
![]() |
(D.46) |
for



Moment Theorem
Theorem:
For a random variable
,
![]() |
(D.47) |
where



![]() |
(D.48) |
(Note that


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

![]() |
(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




![]() |
(D.54) |
In particular,
![\begin{eqnarray*}
\Phi^\prime(\omega) &=& -\frac{1}{2}\sigma^2 2\omega\Phi(\omega)\\ [5pt]
\Phi^{\prime\prime}(\omega) &=& -\frac{1}{2}\sigma^2 2\omega\Phi^\prime(\omega)
-\frac{1}{2}\sigma^2 2\Phi(\omega)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img2865.png)
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