One-dimensional Moments
$
\def \hieruit {\quad \Longrightarrow \quad}
\def \slechts {\quad \Longleftrightarrow \quad}
\def \EN {\quad \mbox{and} \quad}
\def \OF {\quad \mbox{or} \quad}
$
Consider a collection $X$ of arbitrary points $x_k$ in one-dimensional space.
The members of this points cloud can be thought as coordinate positions
on a straight line:
$$
X = \{ x_1 , x_2 , x_3 , ... , x_k , ... , x_{N-1} , x_N \}
$$
A quantity called weight or mass $m_k$ is associated with each of these points.
The total weight or mass $M$ of the points is given by the sum of the partial
weights $m_k$ :
$$
M = \sum_{k=1}^N m_k = \sum_k m_k
$$
It will be assumed in the sequel that the weights are always positive, meaning
that they can be normed. Such normed weights $w_k$ are defined by:
$$
w_k = \frac{m_k}{M} \hieruit 0 \le w_k \le 1 \EN \sum_k w_k = 1
$$
It is remarked that the weights $w_k$ can be interpreted as the components of
a discrete probability distribution. Reason why we are tempted to
conceive a certain spot, called center of mass, center of gravity,
midpoint, middle or simply the mean. It is defined by:
$$
\mu_x = \overline{x} = \sum_k w_k x_k
$$
The midpoint takes a special position at the points cloud, since it is the
weighted mean value of all positions of the points in the points cloud. It's
easy to conceive a weighted mean value of other quantities, however. A most
useful quantity is the so-called second order moment, which is also known
as the moment of inertia, due to its applications in classical mechanics.
Accordingly, the midpoint is also called a first order moment. The second
order moment may also be called (the square of the) standard deviation or
spread, due to the quite analogous quantity in Probability Theory:
$$
\overline{x^2} = \sum_k w_k x_k^2
$$
In addition to the above discrete quantities, there also exist continuum
versions of the first and second order moments. The only difference is that
the latter are defined by (definite) integrals instead of sums:
$$
M = \int_a^b m(x) \, dx = \int m(x) \, dx \EN
w(x) = \frac{m(x)}{M} \hieruit \int w(x) \, dx = 1
$$ $$
\overline{x} = \int w(x) \, x \, dx \EN
\overline{x^2} = \int w(x) \, x^2 \, dx
$$
It is clear from the outset, however, that such integrals are just limiting
cases of discrete sums. Hence subsequent results will also be valid for the
continuous version of the theory.
Second order moments may be defined with respect to a fixed,
but otherwise arbitrary point $p$ in (1-D) space:
$$
\sigma_{xx}(p) = \sum_k w_k (x_k - p)^2 \OF
\sigma_{xx}(p) = \int w(x) \, (x - p)^2 \, dx
$$
The moment of inertia is interpreted as a mean of the squared distances of the
points in the cloud with respect to a fixed point $p$.
It will be shown now that there exists a preferrable origin, which is precisely
the midpoint of the points distribution.
$$
\sigma_{xx}(p) = \sum_k w_k (x_k - p)^2 =
\sum_k w_k x_k^2 - 2 p \sum_k w_k x_k + p^2 =
$$ $$
\overline{x^2} - 2 p \overline{x} + p^2 =
\left[ \overline{x^2} - \overline{x}^2 \right]
+ \left[ \overline{x}^2 - 2 p \overline{x} + p^2 \right]
$$
The first term between square brackets $\left[ \, \right]$ can be worked out
as follows:
$$
\left[ \sum_k w_k x_k^2 - \left( \sum_k w_k x_k \right)^2 \right] =
$$ $$
\sum_k w_k x_k^2 - 2 \sum_k w_k \left( \sum_L w_L x_L \right) x_k
+ \sum_k w_k \left( \sum_L w_L x_L \right)^2 =
$$ $$
\sum_k w_k \left[ x_k^2 - 2 \left( \sum_L w_L x_L \right) x_k
+ \left( \sum_L w_L x_L \right)^2 \right] =
$$ $$
\sum_k w_k \left[ x_k - \left( \sum_L w_L x_L \right) \right]^2 =
\sum_k w_k \left( x_k - \overline{x} \right)^2
$$
And the second term between square brackets $\left[ \, \right]$ is:
$$
\left[ \overline{x}^2 - 2 p \overline{x} + p^2 \right]
= \left( \overline{x} - p \right)^2
$$
Conclusion:
$$
\sum_k w_k (x_k - p)^2 =
\sum_k w_k \left( x_k - \overline{x} \right)^2 +
\left( \overline{x} - p \right)^2
$$
Then we see that the first term is positive, because it is a sum of (weighted)
squares.
But also the second term is a square and hence positive. The latter assumes
a minimum if it is exactly zero, that is if: $ p = \overline{x} $ . Formally:
$$
\sum_{k} w_k (x_k - p)^2 = \mbox{minimum}(p) \slechts
p = \overline{x} = \sum_k w_k x_k
$$
The physical interpretation of the above is that a moment of inertia assumes
a minimal value with respect to the origin if that origin is coincident with
the center of mass. A moment of inertia with respect to an origin which is
different from the center of mass can be expressed as the sum of two moments:
one which expresses the moment of inertia with respect to the midpoint plus
one which expresses the moment of inertia of the midpoint with respect to the
origin. Unless explicitly stated otherwise, it will be assumed in the sequel
that all moments of inertia are defined with respect to the midpoint $\mu_x$
or all (squares of the) spreads with respect to the mean.
Then we can drop the dependence on $(p)$ in:
$$
\sigma_{xx} = \sum_k w_k (x_k - \mu_x)^2
$$