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''of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010330/a0103301.png" />''

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

<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>

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

<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>

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

<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>

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" />.