# Difference between revisions of "Moment"

From Glossary of Meteorology

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#<div class="definition"><div class="short_definition">The product of a distance and another [[parameter]].</div><br/> <div class="paragraph">The moment may be about a point, line, or plane; if the parameter is a [[vector]], the moment is the [[vector product]] of the vector distance from the point, line, or plane, into the parameter. Thus, the moment of the [[momentum]] of a [[fluid parcel]] per unit volume about an axis is '''r''' × ρ'''u''', where '''r''' is the vector from axis to the parcel, ρ the [[density]], and '''u''' the [[velocity]] vector of the parcel; this is also called the [[angular momentum]]. The moment of a force '''F''' about an axis is '''r''' × '''F''', called the [[torque]]. The second moment of a parameter is the moment of the first moment, and so on, for higher moments.</div><br/> </div> | #<div class="definition"><div class="short_definition">The product of a distance and another [[parameter]].</div><br/> <div class="paragraph">The moment may be about a point, line, or plane; if the parameter is a [[vector]], the moment is the [[vector product]] of the vector distance from the point, line, or plane, into the parameter. Thus, the moment of the [[momentum]] of a [[fluid parcel]] per unit volume about an axis is '''r''' × ρ'''u''', where '''r''' is the vector from axis to the parcel, ρ the [[density]], and '''u''' the [[velocity]] vector of the parcel; this is also called the [[angular momentum]]. The moment of a force '''F''' about an axis is '''r''' × '''F''', called the [[torque]]. The second moment of a parameter is the moment of the first moment, and so on, for higher moments.</div><br/> </div> | ||

− | #<div class="definition"><div class="short_definition">By analogy, in [[statistical]] terminology, the [[mean value]] of a power of a [[random variable]].</div><br/> <div class="paragraph">The symbol μ<sub>''n''</sub>′ (or ν<sub>''n''</sub>) is used for a [[raw moment]] as distinguished from the corresponding [[central moment]] μ<sub>''n''</sub> taken about the mean μ. Thus the raw moments are <div class="display-formula"><blockquote>[[File:ams2001glos-Me24.gif|link=|center|ams2001glos-Me24]]</blockquote></div> where ''E''(''x''<sup>''n''</sup>) is the [[expected value]] of the [[variate]] ''x'' to the ''n''th power. In particular, μ<sub>0</sub>′ ≡ 1 and μ<sub>1</sub>′ ≡ ν<sub>1</sub> ≡ μ. The central moments are <div class="display-formula"><blockquote>[[File:ams2001glos-Me25.gif|link=|center|ams2001glos-Me25]]</blockquote></div> where ''E''[(''x'' - μ)<sup>''n''</sup>] is the expected value of the ''n''th power of the [[deviation]] of the variate from its mean. In particular, μ<sub>0</sub> ≡ 1, μ<sub>1</sub> ≡ 0, μ<sub>2</sub> ≡ σ<sup>2</sup>, where σ<sup>2</sup> is the [[variance]].</div><br/> </div> | + | #<div class="definition"><div class="short_definition">By analogy, in [[statistical]] terminology, the [[mean value|mean value]] of a power of a [[random variable]].</div><br/> <div class="paragraph">The symbol μ<sub>''n''</sub>′ (or ν<sub>''n''</sub>) is used for a [[raw moment]] as distinguished from the corresponding [[central moment]] μ<sub>''n''</sub> taken about the mean μ. Thus the raw moments are <div class="display-formula"><blockquote>[[File:ams2001glos-Me24.gif|link=|center|ams2001glos-Me24]]</blockquote></div> where ''E''(''x''<sup>''n''</sup>) is the [[expected value]] of the [[variate]] ''x'' to the ''n''th power. In particular, μ<sub>0</sub>′ ≡ 1 and μ<sub>1</sub>′ ≡ ν<sub>1</sub> ≡ μ. The central moments are <div class="display-formula"><blockquote>[[File:ams2001glos-Me25.gif|link=|center|ams2001glos-Me25]]</blockquote></div> where ''E''[(''x'' - μ)<sup>''n''</sup>] is the expected value of the ''n''th power of the [[deviation]] of the variate from its mean. In particular, μ<sub>0</sub> ≡ 1, μ<sub>1</sub> ≡ 0, μ<sub>2</sub> ≡ σ<sup>2</sup>, where σ<sup>2</sup> is the [[variance]].</div><br/> </div> |

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## Latest revision as of 17:27, 25 April 2012

## moment

- The product of a distance and another parameter.

The moment may be about a point, line, or plane; if the parameter is a vector, the moment is the vector product of the vector distance from the point, line, or plane, into the parameter. Thus, the moment of the momentum of a fluid parcel per unit volume about an axis is**r**× ρ**u**, where**r**is the vector from axis to the parcel, ρ the density, and**u**the velocity vector of the parcel; this is also called the angular momentum. The moment of a force**F**about an axis is**r**×**F**, called the torque. The second moment of a parameter is the moment of the first moment, and so on, for higher moments.

The symbol μ_{n}′ (or ν_{n}) is used for a raw moment as distinguished from the corresponding central moment μ_{n}taken about the mean μ. Thus the raw moments arewhere*E*(*x*^{n}) is the expected value of the variate*x*to the*n*th power. In particular, μ_{0}′ ≡ 1 and μ_{1}′ ≡ ν_{1}≡ μ. The central moments arewhere*E*[(*x*- μ)^{n}] is the expected value of the*n*th power of the deviation of the variate from its mean. In particular, μ_{0}≡ 1, μ_{1}≡ 0, μ_{2}≡ σ^{2}, where σ^{2}is the variance.