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''of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103301.png" />''
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The mathematical expectation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103303.png" />. It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103304.png" />, so that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103305.png" /></td> </tr></table>
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''of a random variable  $  X $''
  
The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103306.png" /> is called the order of the absolute moment. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103307.png" /> is the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103308.png" />, then
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The mathematical expectation of  $  | X |  ^ {r} $,
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$  r > 0 $.  
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It is usually denoted by  $  \beta _ {r} $,  
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so that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103309.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$
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\beta _ {r}  = {\mathsf E} | X |  ^ {r} .
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$$
  
and, for example, if the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033010.png" /> has density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033011.png" />, one has
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The number  $  r $
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is called the order of the absolute moment. If  $  F (x) $
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is the distribution function of $  X $,  
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then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
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\beta _ {r}  = \int\limits _ {- \infty } ^ { {+ }  \infty } | x |  ^ {r}  d F ( x ) ,
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$$
  
In relation to the equations (1) and (2) one also speaks, respectively, of the absolute moments of the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033013.png" /> and the density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033014.png" />. The existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033015.png" /> implies the existence of the absolute moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033016.png" /> and also of the moments (cf. [[Moment|Moment]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033017.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033018.png" />. Absolute moments often appear in estimates of probability distributions and their characteristic functions (cf. [[Chebyshev inequality in probability theory|Chebyshev inequality in probability theory]]; [[Lyapunov theorem|Lyapunov theorem]]). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033019.png" /> is a convex function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033020.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033021.png" /> is a non-decreasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a01033023.png" />.
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and, for example, if the distribution of  $  X $
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has density  $  p (x) $,
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one has
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$$ \tag{2 }
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\beta _ {r}  =  \int\limits _ {- \infty } ^ { {+ }  \infty }
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| x |  ^ {r} p ( x )  dx .
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$$
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In relation to the equations (1) and (2) one also speaks, respectively, of the absolute moments of the distribution function $  F(x) $
 +
and the density $  p(x) $.  
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The existence of $  \beta _ {r} $
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implies the existence of the absolute moment $  \beta _ {r  ^  \prime  } $
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and also of the moments (cf. [[Moment|Moment]]) of order $  r  ^  \prime  $,  
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for $  0 < r  ^  \prime  \leq  r $.  
 +
Absolute moments often appear in estimates of probability distributions and their characteristic functions (cf. [[Chebyshev inequality in probability theory|Chebyshev inequality in probability theory]]; [[Lyapunov theorem|Lyapunov theorem]]). The function $  \mathop{\rm log}  \beta _ {r} $
 +
is a convex function of $  r $,  
 +
and the function $  \beta _ {r}  ^ {1/r} $
 +
is a non-decreasing function of $  r $,
 +
$  r > 0 $.

Latest revision as of 16:08, 1 April 2020


of a random variable $ X $

The mathematical expectation of $ | X | ^ {r} $, $ r > 0 $. It is usually denoted by $ \beta _ {r} $, so that

$$ \beta _ {r} = {\mathsf E} | X | ^ {r} . $$

The number $ r $ is called the order of the absolute moment. If $ F (x) $ is the distribution function of $ X $, then

$$ \tag{1 } \beta _ {r} = \int\limits _ {- \infty } ^ { {+ } \infty } | x | ^ {r} d F ( x ) , $$

and, for example, if the distribution of $ X $ has density $ p (x) $, one has

$$ \tag{2 } \beta _ {r} = \int\limits _ {- \infty } ^ { {+ } \infty } | x | ^ {r} p ( x ) dx . $$

In relation to the equations (1) and (2) one also speaks, respectively, of the absolute moments of the distribution function $ F(x) $ and the density $ p(x) $. The existence of $ \beta _ {r} $ implies the existence of the absolute moment $ \beta _ {r ^ \prime } $ and also of the moments (cf. Moment) of order $ r ^ \prime $, for $ 0 < r ^ \prime \leq r $. Absolute moments often appear in estimates of probability distributions and their characteristic functions (cf. Chebyshev inequality in probability theory; Lyapunov theorem). The function $ \mathop{\rm log} \beta _ {r} $ is a convex function of $ r $, and the function $ \beta _ {r} ^ {1/r} $ is a non-decreasing function of $ r $, $ r > 0 $.

How to Cite This Entry:
Absolute moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_moment&oldid=45005
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article