Moment (mathematics)

"Second moment" redirects here. For the technique in probability theory, see Second moment method.See also: Moment (physics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" (in a particular sense) of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by an ellipsoid. Other moments describe other aspects of a distribution such as how the distribution is skewed from its mean, or peaked. The mathematical concept is closely related to the concept of moment in physics, although moment in physics is often represented somewhat differently. Any distribution can be characterized by a number of features (such as the mean, the variance, the skewness, etc.), and the moments of a function^{[1]} describe the nature of its distribution.
The 1st moment is denoted by μ_{1}. The first moment of the distribution of the random variable X is the expectation operator, i.e., the population mean (if the first moment exists).
In higher orders, the central moments (moments about the mean) are more interesting than the moments about zero. The kth central moment, of a realvalued random variable probability distribution X, with the expected value μ is:
The first central moment is thus 0. The zeroth central moment, μ_{0} is one. See also central moment.
Other moments may also be defined. For example, the n th inverse moment about zero is E(X ^{− n}) and the n th logarithmic moment about zero is E(ln ^{n}(x)).
Contents
Significance of the moments
The n^{th} moment of a realvalued continuous function f(x) of a real variable about a value c is
It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces. The moment of a function, without further explanation, usually refers to the above expression with c = 0.
Usually, except in the special context of the problem of moments, the function f(x) will be a probability density function. The n^{th} moment about zero of a probability density function f(x) is the expected value of X^{n} and is called a raw moment or crude moment.^{[2]} The moments about its mean μ are called central moments; these describe the shape of the function, independently of translation.
If f is a probability density function, then the value of the integral above is called the nth moment of the probability distribution. More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the nth moment of the probability distribution is given by the Riemann–Stieltjes integral
where X is a random variable that has this distribution and E the expectation operator or mean.
When
then the moment is said not to exist. If the nth moment about any point exists, so does (n − 1)th moment, and all lowerorder moments, about every point.
Variance
Main article: VarianceThe second central moment about the mean is the variance. Its positive square root is the standard deviation σ.
Normalized moments
The normalized nth central moment or standardized moment is the nth central moment divided by σ^{n}; the normalized nth central moment of x = E((x − μ)^{n})/σ^{n}. These normalized central moments are dimensionless quantities, which represent the distribution independently of any linear change of scale.
Skewness
Main article: SkewnessThe third central moment is a measure of the lopsidedness of the distribution; any symmetric distribution will have a third central moment, if defined, of zero. The normalized third central moment is called the skewness, often γ. A distribution that is skewed to the left (the tail of the distribution is heavier on the left) will have a negative skewness. A distribution that is skewed to the right (the tail of the distribution is heavier on the right), will have a positive skewness.
For distributions that are not too different from the normal distribution, the median will be somewhere near μ − γσ/6; the mode about μ − γσ/2.
Kurtosis
Main article: KurtosisThe fourth central moment is a measure of whether the distribution is tall and skinny or short and squat, compared to the normal distribution of the same variance. Since it is the expectation of a fourth power, the fourth central moment, where defined, is always nonnegative; and except for a point distribution, it is always strictly positive. The fourth central moment of a normal distribution is 3σ^{4}.
The kurtosis κ is defined to be the normalized fourth central moment minus 3. (Equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance.) Some authorities^{[3]}^{[4]} do not subtract three, but it is usually more convenient to have the normal distribution at the origin of coordinates. If a distribution has a peak at the mean and long tails, the fourth moment will be high and the kurtosis positive (leptokurtic); and conversely; thus, bounded distributions tend to have low kurtosis (platykurtic).
The kurtosis can be positive without limit, but κ must be greater than or equal to γ^{2} − 2; equality only holds for binary distributions. For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of γ^{2} and 2γ^{2}.
The inequality can be proven by considering
where T = (X − μ)/σ. This is the expectation of a square, so it is nonnegative whatever a is; on the other hand, it's also a quadratic equation in a. Its discriminant must be nonpositive, which gives the required relationship.
Mixed moments
Mixed moments are moments involving multiple variables.
Some examples are covariance, coskewness and cokurtosis. While there is a unique covariance, there are multiple coskewnesses and cokurtoses.
Higher moments
Highorder moments are moments beyond 4thorder moments. The higher the moment, the harder it is to estimate, in the sense that larger samples are required in order to obtain estimates of similar quality.^{[citation needed]}
Cumulants

Main article: cumulant
The first moment and the second and third unnormalized central moments are additive in the sense that if X and Y are independent random variables then
and
and
(These can also hold for variables that satisfy weaker conditions than independence. The first always holds; if the second holds, the variables are called uncorrelated).
In fact, these are the first three cumulants and all cumulants share this additivity property.
Sample moments
The moments of a population can be estimated using the sample kth moment
applied to a sample X_{1},X_{2},..., X_{n} drawn from the population.
It can be shown that the expected value of the sample moment is equal to the kth moment of the population, if that moment exists, for any sample size n. It is thus an unbiased estimator.
Problem of moments
Main article: moment problemThe problem of moments seeks characterizations of sequences { μ′_{n} : n = 1, 2, 3, ... } that are sequences of moments of some function f.
Partial moments
Partial moments are sometimes referred to as "onesided moments." The nth order lower and upper partial moments with respect to a reference point r may be expressed as
Partial moments are normalized by being raised to the power 1/n. The upside potential ratio may be expressed as a ratio of a firstorder upper partial moment to a normalized secondorder lower partial moment.
Moments in metric spaces
Let (M, d) be a metric space, and let B(M) be the Borel σalgebra on M, the σalgebra generated by the dopen subsets of M. (For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ +∞.
The p^{th} moment of a measure μ on the measurable space (M, B(M)) about a given point x_{0} in M is defined to be
μ is said to have finite p^{th} moment if the p^{th} moment of μ about x_{0} is finite for some x_{0} ∈ M.
This terminology for measures carries over to random variables in the usual way: if (Ω, Σ, P) is a probability space and X : Ω → M is a random variable, then the p^{th} moment of X about x_{0} ∈ M is defined to be
and X has finite p^{th} moment if the p^{th} moment of X about x_{0} is finite for some x_{0} ∈ M.
See also
 Hamburger moment problem
 Hausdorff moment problem
 Image moments
 Lmoment
 Method of moments
 Second moment method
 Standardized moment
 Stieltjes moment problem
 Taylor expansions for the moments of functions of random variables
References
 ^ A function such as a probability density function or cumulative distribution function; see Momentgenerating function.
 ^ http://mathworld.wolfram.com/RawMoment.html Raw Moments at Mathworld
 ^ Casella, George; Berger, Roger L. (2002). Statistical Inference (2 ed.). Pacific Grove: Duxbury. ISBN 0534243126.
 ^ Ballanda, Kevin P.; MacGillivray, H. L. (1988). "Kurtosis: A Critical Review". The American Statistician (American Statistical Association) 42 (2): 111–119. doi:10.2307/2684482. JSTOR 2684482.
External links
Theory of probability distributions probability mass function (pmf) · probability density function (pdf) · cumulative distribution function (cdf) · quantile functionCategories: Probability theory
 Mathematical analysis
 Theory of probability distributions

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