Difference between revisions of "Moment"
(Importing text file) |
(→References: Feller: internal link) |
||
Line 33: | Line 33: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''2''', Wiley (1971) Chapt. 1</TD></TR></table> |
Revision as of 11:17, 4 May 2012
A numerical characteristic of a probability distribution. The moment of order (
an integer) of a random variable
is defined as the mathematical expectation
, if it exists. If
is the distribution function of the random variable
, then
![]() | (*) |
For the definition of a moment in probability theory, a direct analogy is used with the corresponding idea which plays a major role in mechanics: Formula (*) is defined as the moment of a mass distribution. The first-order moment (a statistical moment in mechanics) of a random variable is the mathematical expectation
. The value
is called the moment of order
relative to
,
is the central moment of order
. The second-order central moment
is called the dispersion (or variance)
(the moment of inertia in mechanics). The value
is called the absolute moment of order
(absolute moments are also defined for non-integral
). The moments of the joint distribution of random variables
(see Multi-dimensional distribution) are defined similarly: For any integers
,
, the mathematical expectation
is called a mixed moment of order
, and
is called a central mixed moment of order
. The mixed moment
is called the covariance and is one of the basic characteristics of dependency between random variables (see Correlation (in statistics)). Many properties of moments (in particular, the inequality for moments) are consequences of the fact that for any random variable
the function
is convex with respect to
in each finite interval on which the function is defined;
is a non-decreasing function of
. The moments
and
exist if and only if
. The existence of
implies the existence of all moments of orders
. If
for all
, then the mixed moments
exist for all
,
. In some cases, for the definition of moments, the so-called moment generating function is useful — the function
with the moments of the distribution as coefficients in its power-series expansion; for integer-valued random variables this function is related to the generating function
by the relation
. If
, then the characteristic function
of the random variable
has continuous derivatives up to order
inclusively, and the moment of order
is the coefficient of
in the expansion of
in powers of
,
![]() |
If the characteristic function has a derivative of order at zero, then
.
For the connection of moments with semi-invariants see Semi-invariant. If the moments of a distribution are known, then it is possible to make some assertions about the probabilities of deviation of a random variable from its mathematical expectation in terms of inequalities; the best known are the Chebyshev inequality in probability theory
![]() |
and its generalizations.
Problems of determining a probability distribution from its sequence of moments are called moment problems (cf. Moment problem). Such problems were first discussed by P.L. Chebyshev (1874) in connection with research on limit theorems. In order that the probability distribution of a random variable be uniquely defined by its moments
it is sufficient, for example, that Carleman's condition be satisfied:
![]() |
A similar result even holds for moments of random vectors.
The use of moments in the proof of limit theorems is based on the following fact. Let ,
be a sequence of distribution functions, all moments
of which are finite, and for each integer
let
![]() |
where is finite. Then there is a sequence
that weakly converges to a distribution function
having
as its moments. If the moments determine
uniquely, then the sequence
weakly converges to
. Based on this is the so-called method of moments (cf. Moments, method of (in probability theory)), used, in particular, in mathematical statistics for the study of the deviation of empirical distributions from theoretical ones, and for the statistical estimation of parameters of a distribution (on sampling moments as well as estimators for moments of a certain distribution see Empirical distribution).
Comments
The moment generating function is defined by .
References
[a1] | W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) Chapt. 1 |
Moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moment&oldid=25957