Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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 ,