Moment (mathematics)

momentsmomentraw momentfirst momentmoment methodmoments in metric spacesraw momentssecond momentvanishing momentsabsolute moment
In mathematics, a moment is a specific quantitative measure of the shape of a function.wikipedia
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Moment of inertia

rotational inertiamoments of inertiainertia tensor
If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia.
Its simplest definition is the second moment of mass with respect to distance from an axis.

Central moment

central momentsmoment about the meanmoments about the mean
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.
In probability theory and statistics, a central moment is a moment of a probability distribution of a random variable about the random variable's mean; that is, it is the expected value of a specified integer power of the deviation of the random variable from the mean.

Kurtosis

excess kurtosisleptokurticplatykurtic
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The kurtosis is defined to be the normalised fourth central moment minus 3 (Equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance).
The standard measure of kurtosis, originating with Karl Pearson, is based on a scaled version of the fourth moment of the data or population.

Hausdorff moment problem

completely monotone sequenceHausdorff
) uniquely determines the distribution (Hausdorff moment problem).
be the sequence of moments

Hamburger moment problem

The same is not true on unbounded intervals (Hamburger moment problem).
In other words, an affirmative answer to the problem means that { m n : n = 0, 1, 2, ... } is the sequence of moments of some positive Borel measure μ.

Standardized moment

standardized central moments
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.
Let X be a random variable with a probability distribution P and mean value (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of X.

Variance

sample variancepopulation variancevariability
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.
The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. . Conversely, if a continuous function \varphi satisfies for all random variables X, then it is necessarily of the form, where a > 0.

Random variable

random variablesrandom variationrandom
It is possible to define moments for random variables in a more general fashion than moments for real values—see moments in metric spaces.
In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.

Cumulant

cumulant generating functioncumulantscumulant-generating function
The kurtosis is defined to be the normalised fourth central moment minus 3 (Equivalently, as in the next section, it is the fourth cumulant divided by the square of the variance).
In probability theory and statistics, the cumulants of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution.

Skewness

skewedskewskewed distribution
If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis.
where is the sample mean, s is the sample standard deviation, and the numerator m 3 is the sample third central moment.

Normal distribution

normally distributednormalGaussian
For distributions that are not too different from the normal distribution, the median will be somewhere near
The plain and absolute moments of a variable X are the expected values of X^p and |X|^p, respectively.

Moment (physics)

momentmomentsmoment arm
The mathematical concept is closely related to the concept of moment in physics.
The concept of moment in physics is derived from the mathematical concept of moments.

Higher-order statistics

high-orderhigh-order statisticshigher order statistics
As with variance, skewness, and kurtosis, these are higher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters.
The third and higher moments, as used in the skewness and kurtosis, are examples of HOS, whereas the first and second moments, as used in the arithmetic mean (first), and variance (second) are examples of low-order statistics.

Shape parameter

shape
As with variance, skewness, and kurtosis, these are higher-order statistics, involving non-linear combinations of the data, and can be used for description or estimation of further shape parameters.
Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment), if the higher moments are defined and finite.

Standard deviation

standard deviationssample standard deviationsigma
The positive square root of the variance is the standard deviation
The calculation of the sum of squared deviations can be related to moments calculated directly from the data.

Expected value

expectationexpectedmean
The -th moment about zero of a probability density function f(x) is the expected value of and is called a raw moment or crude moment.
The expected values of the powers of X are called the moments of X; the moments about the mean of X are expected values of powers of X − E[X]. The moments of some random variables can be used to specify their distributions, via their moment generating functions.

Riemann–Stieltjes integral

Riemann–StieltjesRiemann-Stieltjes integrationRiemann–Stieltjes integral application to probability theory
More generally, if F is a cumulative probability distribution function of any probability distribution, which may not have a density function, then the -th moment of the probability distribution is given by the Riemann–Stieltjes integral
In particular, no matter how ill-behaved the cumulative distribution function g of a random variable X, if the moment E(X n ) exists, then it is equal to

Upside potential ratio

upside-potential ratio
The upside potential ratio may be expressed as a ratio of a first-order upper partial moment to a normalized second-order lower partial moment.
The upside-potential ratio may also be expressed as a ratio of partial moments since is the first upper moment and is the second lower partial moment.

L-moment

probability weighted moments
L-moment
They are linear combinations of order statistics (L-statistics) analogous to conventional moments, and can be used to calculate quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively (the L-mean is identical to the conventional mean).

Image moment

geometric momentsimage momentsMoment invariant
Image moment
In image processing, computer vision and related fields, an image moment is a certain particular weighted average (moment) of the image pixels' intensities, or a function of such moments, usually chosen to have some attractive property or interpretation.

Correlation and dependence

correlationcorrelatedcorrelate
The first always holds; if the second holds, the variables are called uncorrelated).
For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined.

Method of moments (statistics)

method of momentsmethod of matching momentsmethod of moment matching
Method of moments (statistics)
One starts with deriving equations that relate the population moments (i.e., the expected values of powers of the random variable under consideration) to the parameters of interest.

Method of moments (probability theory)

Method of moments
Method of moments (probability theory)
In probability theory, the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences.

Moment-generating function

moment generating functionCalculations of momentsgenerating functions
Moment-generating function
where m_n is the nth moment.