30.5 PROPERTIES OF DISTRIBUTIONS
We note that the notationμkandνkfor the moments and central moments
respectively is not universal. Indeed, in some books their meanings are reversed.
We can write thekth central moment of a distribution in terms of itskth and
lower-order moments by expanding (X−μ)kin powers ofX. We have already
noted thatν 2 =μ 2 −μ^21 , and similar expressions may be obtained for higher-order
central moments. For example,
ν 3 =E
[
(X−μ 1 )^3
]
=E
[
X^3 − 3 μ 1 X^2 +3μ^21 X−μ^31
]
=μ 3 − 3 μ 1 μ 2 +3μ^21 μ 1 −μ^31
=μ 3 − 3 μ 1 μ 2 +2μ^31. (30.53)
In general, it is straightforward to show that
νk=μk−kC 1 μk− 1 μ 1 +···+(−1)rkCrμk−rμr 1 +···+(−1)k−^1 (kCk− 1 −1)μk 1.
(30.54)
Once again, direct evaluation of the sum or integral in (30.52) can be rather
tedious for higher moments, and it is usually quicker to use the moment generating
function (see subsection 30.7.2), from which the central moments can be easily
evaluated as well.
The PDF for a Gaussian distribution (see subsection 30.9.1) with meanμand variance
σ^2 is given by
f(x)=
1
σ
√
2 π
exp
[
−
(x−μ)^2
2 σ^2
]
.
Obtain an expression for thekth central moment of this distribution.
As an illustration, we will perform this calculation by evaluating the integral in (30.52)
directly. Thus, thekth central moment off(x)isgivenby
νk=
∫∞
−∞
(x−μ)kf(x)dx
=
1
σ
√
2 π
∫∞
−∞
(x−μ)kexp
[
−
(x−μ)^2
2 σ^2
]
dx
=
1
σ
√
2 π
∫∞
−∞
ykexp
(
−
y^2
2 σ^2
)
dy, (30.55)
where in the last line we have made the substitutiony=x−μ. It is clear that ifkis
odd then the integrand is an odd function ofyand hence the integral equals zero. Thus,
νk=0ifkis odd. Whenkis even, we could calculateνkby integrating by parts to obtain
a reduction formula, but it is more elegant to consider instead the standard integral (see
subsection 6.4.2)
I=
∫∞
−∞
exp(−αy^2 )dy=π^1 /^2 α−^1 /^2 ,