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| {{redirect|Second moment|the technique in probability theory|Second moment method}}
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| {{see also|Moment (physics)}}
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| In [[mathematics]], a '''moment''' is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", or more specifically the "second [[central 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 (mathematics)|distribution]] such as how the distribution is skewed from its mean. The mathematical concept is closely related to the concept of [[moment (physics)|moment]] in [[physics]], although moment in physics is [[Moment_of_inertia#Comparison_with_covariance_matrix|often represented somewhat differently]].{{Dead link|date=October 2013}} Any distribution can be characterized by a number of features (such as the [[mean]], the [[variance]], the [[skewness]], etc.), and the moments of a random variable's [[probability distribution]] are related to these features. The probability distribution itself can be expressed as a [[probability density function]], [[probability mass function]], [[cumulative distribution function]], [[characteristic function (probability)|characteristic function]], or [[moment-generating function]].
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| The first '''raw moment''', or first moment about zero or simply the first moment, is referred to as the distribution's mean. The mean of the distribution of the random variable ''X'', if the mean exists, is referred to with the [[Expected value|expectation operator]].
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| In higher orders, the '''central moments''' (moments about the mean) are more interesting than the moments about zero, because they provide clearer information about the distribution's shape.
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| Other moments may also be defined. For example, the ''n''th inverse moment about zero is <math>E(X^{-n})</math> and the ''n'' th logarithmic moment about zero is <math>E(\ln^n(X))</math>.
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| ==Significance of the moments==
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| The ''n''th moment of a real-valued continuous function ''f''(''x'') of a real variable about a value ''c'' is
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| :<math>\mu'_n=\int_{-\infty}^\infty (x - c)^n\,f(x)\,dx.\,\!</math>
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| It is possible to define moments for [[random variable]]s in a more general fashion than moments for real values—see [[Moment_(mathematics)#Central_Moments_in_metric_spaces|moments in metric spaces]]. The moment of a function, without further explanation, usually refers to the above expression with ''c'' = 0.
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| Usually, except in the special context of the [[Moment problem|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''<sup>''n''</sup> and is called a ''raw moment'' or ''crude moment''.<ref>http://mathworld.wolfram.com/RawMoment.html Raw Moments at Math-world</ref> The moments about its mean μ are called [[central moment|''central'' moments]]; these describe the shape of the function, independently of [[translation (geometry)|translation]].
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| If ''f'' is a [[probability density function]], then the value of the integral above is called the ''n''th moment of the [[probability distribution]]. More generally, if ''F'' is a [[cumulative distribution function|cumulative probability distribution function]] of any probability distribution, which may not have a density function, then the ''n''th moment of the probability distribution is given by the [[Riemann–Stieltjes integral]]
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| :<math>\mu'_n = \operatorname{E}(X^n)=\int_{-\infty}^\infty x^n\,dF(x)\,</math>
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| where ''X'' is a [[random variable]] that has this cumulative distribution ''F'', and '''E''' is the [[expectation operator]] or mean.
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| When
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| :<math>\operatorname{E}(|X^n|) = \int_{-\infty}^\infty |x^n|\,dF(x) = \infty,\,</math>
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| then the moment is said not to exist. If the ''n''th moment about any point exists, so does (''n'' − 1)th moment (and thus, all lower-order moments) about every point.
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| The zeroth moment of any [[probability density function]] is 1, since the area under any [[probability density function]] must be equal to one.
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| {|class="wikitable"
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| |+Significance of moments (raw, central, standardized) and cumulants (raw, standardized), in connection with named properties of distributions
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| ! Moment number !! Raw moment !! Central moment !! Standardized moment !! Raw cumulant !! Standardized cumulant
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| |-
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| | 1 || [[mean]] || 0 || 0 || [[mean]] || N/A
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| |-
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| | 2 || – || [[variance]] || 1 || [[variance]] || 1
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| |-
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| | 3 || – || – || [[skewness]] || – || [[skewness]]
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| |-
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| | 4 || – || – || historical [[kurtosis]] (or flatness) || – || modern [[kurtosis]] (i.e. excess kurtosis)
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| |-
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| | 5 || – || – || hyperskewness || – || –
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| |-
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| | 6 || – || – || hyperflatness || – || –
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| |-
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| | 7+ || – || – || - || – || –
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| |}
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| ===Mean===
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| {{Main|Mean}}
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| The first raw moment is the [[mean]].
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| ===Variance===
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| {{Main|Variance}}
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| The second [[central moment]] is the [[variance]]. Its positive square root is the [[standard deviation]] ''σ''.
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| ==== Normalized moments ====
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| The ''normalized'' ''n''th central moment or [[standardized moment]] is the ''n''th central moment divided by ''σ''<sup>''n''</sup>; the normalized ''n''th central moment of <math>x = E((x - \mu)^n)/ \sigma^n</math>. These normalized central moments are [[dimensionless number|dimensionless quantities]], which represent the distribution independently of any linear change of scale.
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| ===Skewness===
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| {{Main|Skewness}}
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| The 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.
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| For distributions that are not too different from the [[normal distribution]], the [[median]] will be somewhere near μ − γσ/6; the [[Mode (statistics)|mode]] about μ − γσ/2.
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| ===Kurtosis===
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| {{Main|Kurtosis}}
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| The 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 positive; and except for a [[degenerate probability distribution|point distribution]], it is always strictly positive. The fourth central moment of a normal distribution is 3σ<sup>4</sup>.
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| 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<ref name="CasellaBerger">{{cite book
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| | last1 = Casella
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| | first1 = George
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| | last2 = Berger
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| | first2 = Roger L.
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| | authorlink1 = George Casella
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| | authorlink2 = Roger L. Berger
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| | title = Statistical Inference
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| | publisher = [[Duxbury]]
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| | location = Pacific Grove
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| | year = 2002
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| | edition = 2
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| | isbn = 0-534-24312-6 }}</ref><ref name="BalandaMacGillivray88">{{cite journal
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| | last1 = Ballanda
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| | first1 = Kevin P.
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| | last2 = MacGillivray
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| | first2 = H. L.
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| | title = Kurtosis: A Critical Review
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| | journal = The American Statistician
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| | volume = 42
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| | issue = 2
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| | pages = 111–119
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| | year = 1988
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| | doi = 10.2307/2684482
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| | jstor = 2684482
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| | publisher = American Statistical Association}}</ref> 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); conversely, bounded distributions tend to have low kurtosis (platykurtic).
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| The kurtosis can be positive without limit, but κ must be greater than or equal to γ<sup>2</sup> − 2; equality only holds for [[Bernoulli distribution|binary distributions]]. For unbounded skew distributions not too far from normal, κ tends to be somewhere in the area of γ<sup>2</sup> and 2γ<sup>2</sup>.
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| The inequality can be proven by considering
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| :<math>\operatorname{E} ((T^2 - aT - 1)^2)\,</math>
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| where ''T'' = (''X'' − μ)/σ. This is the expectation of a square, so it is non-negative for all ''a''; however it is also a quadratic [[polynomial]] in ''a''. Its [[discriminant]] must be non-positive, which gives the required relationship.
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| === Mixed moments ===
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| '''Mixed moments''' are moments involving multiple variables.
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| Some examples are [[covariance]], [[coskewness]] and [[cokurtosis]]. While there is a unique covariance, there are multiple co-skewnesses and co-kurtoses.
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| === Higher moments ===
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| '''High-order moments''' are moments beyond 4th-order moments. 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 parameter]]s. 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. This is due to the excess [[Degrees_of_freedom_(statistics)|degrees of freedom]] consumed by the higher orders. Further, they can be subtle to interpret, often being most easily understood in terms of lower order moments – compare the higher derivatives of [[Jerk (physics)|jerk]] and [[jounce]] in [[physics]]. For example, just as the 4th-order moment (kurtosis) can be interpreted as "relative importance of tails versus shoulders in causing dispersion" (for a given dispersion, high kurtosis corresponds to heavy tails, while low kurtosis corresponds to heavy shoulders), the 5th-order moment can be interpreted as measuring "relative importance of tails versus center (mode, shoulders) in causing skew" (for a given skew, high 5th moment corresponds to heavy tail and little movement of mode, while low 5th moment corresponds to more change in shoulders).
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| ==Cumulants==
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| {{main|cumulant}}
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| The first moment and the second and third ''unnormalized central'' moments are additive in the sense that if ''X'' and ''Y'' are [[statistical independence|independent]] random variables then
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| :<math>\mu_1(X+Y)=\mu_1(X)+\mu_1(Y)\,</math>
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| and
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| :<math>\operatorname{Var}(X+Y)=\operatorname{Var}(X) + \operatorname{Var}(Y)</math>
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| and
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| :<math>\mu_3(X+Y)=\mu_3(X)+\mu_3(Y).\,</math>
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| (These can also hold for variables that satisfy weaker conditions than independence. The first always holds; if the second holds, the variables are called [[correlation|uncorrelated]]).
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| In fact, these are the first three cumulants and all cumulants share this additivity property.
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| == Sample moments ==
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| For all ''k'', the ''k''-th raw moment of a population can be estimated using the ''k''-th raw sample moment
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| :<math>\frac{1}{n}\sum_{i = 1}^{n} X^k_i\,\!</math>
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| applied to a sample ''X''<sub>1</sub>,''X''<sub>2</sub>,..., ''X''<sub>''n''</sub> drawn from the population.
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| It can be shown that the expected value of the raw sample moment is equal to the ''k''-th raw moment of the population, if that moment exists, for any sample size ''n''. It is thus an unbiased estimator. This contrasts with the situation for central moments, whose computation uses up a degree of freedom by using the sample mean. So for example an unbiased estimate of the population variance (the second central moment) is given by
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| :<math>\frac{1}{n-1}\sum_{i = 1}^{n} (X_i-\bar X)^2\,\!</math>
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| in which the previous denominator ''n'' has been replaced by the degrees of freedom ''n''−1, and in which <math>\bar X</math> refers to the sample mean. This estimate of the population moment is greater than the unadjusted observed sample moment by a factor of <math>\tfrac{n}{n-1},</math> and it is referred to as the "adjusted sample variance" or sometimes simply the "sample variance".
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| ==Problem of moments==
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| {{main|Moment problem}}
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| The ''problem of moments'' seeks characterizations of sequences { ''μ''′<sub>''n''</sub> : ''n'' = 1, 2, 3, ... } that are sequences of moments of some function ''f''.
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| ==Partial moments==
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| Partial moments are sometimes referred to as "one-sided moments." The ''n''th order lower and upper partial moments with respect to a reference point ''r'' may be expressed as
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| :<math>\mu_n^-(r)=\int_{-\infty}^r (r - x)^n\,f(x)\,dx,</math>
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| :<math>\mu_n^+(r)=\int_r^\infty (x - r)^n\,f(x)\,dx.</math>
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| Partial moments are normalized by being raised to the power 1/''n''. 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. They have been used in the definition of some financial metrics, such as the [[Sortino ratio]], as they focus purely on upside or downside.
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| ==Central moments in metric spaces==
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| Let (''M'', ''d'') be a [[metric space]], and let B(''M'') be the [[Borel sigma algebra|Borel σ-algebra]] on ''M'', the [[sigma algebra|σ-algebra]] generated by the ''d''-[[open set|open subsets]] of ''M''. (For technical reasons, it is also convenient to assume that ''M'' is a [[separable space]] with respect to the [[metric (mathematics)|metric]] ''d''.) Let 1 ≤ ''p'' ≤ +∞.
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| The '''''p''th central moment''' of a measure ''μ'' on the [[measurable space]] (''M'', B(''M'')) about a given point ''x''<sub>0</sub> in ''M'' is defined to be
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| :<math>\int_{M} d(x, x_{0})^{p} \, \mathrm{d} \mu (x).</math>
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| ''μ'' is said to have '''finite ''p''th central moment''' if the ''p''th central moment of ''μ'' about ''x''<sub>0</sub> is finite for some ''x''<sub>0</sub> ∈ ''M''.
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| 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 central moment''' of ''X'' about ''x''<sub>0</sub> ∈ ''M'' is defined to be
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| :<math>\int_{M} d (x, x_{0})^{p} \, \mathrm{d} \left( X_{*} (\mathbf{P}) \right) (x) \equiv \int_{\Omega} d (X(\omega), x_{0})^{p} \, \mathrm{d} \mathbf{P} (\omega),</math>
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| and ''X'' has '''finite ''p''th central moment''' if the ''p''th central moment of ''X'' about ''x''<sub>0</sub> is finite for some ''x''<sub>0</sub> ∈ ''M''.
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| ==See also==
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| * [[Factorial moment]]
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| * [[Generalized mean]]
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| * [[Hamburger moment problem]]
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| * [[Hausdorff moment problem]]
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| * [[Image moments]]
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| * [[L-moment]]
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| * [[Method of moments (probability theory)]]
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| * [[Method of moments (statistics)]]
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| * [[Moment-generating function#Calculations of moments|Moment-generating function]]
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| * [[Moment measure]]
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| * [[Second moment method]]
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| * [[Standardized moment]]
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| * [[Stieltjes moment problem]]
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| * [[Taylor expansions for the moments of functions of random variables]]
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| == References ==
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| <references/>
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| ==External links==
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| *{{springer|title=Moment|id=p/m064580}}
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| *[http://mathworld.wolfram.com/topics/Moments.html Moments at Mathworld]
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| *[http://www.geo.upm.es/postgrado/CarlosLopez/geost_03/node37.html Higher Moments]
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| {{Theory of probability distributions}}
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| {{Statistics|descriptive}}
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| {{DEFAULTSORT:Moment (Mathematics)}}
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| [[Category:Probability theory]]
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| [[Category:Mathematical analysis]]
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| [[Category:Theory of probability distributions]]
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