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Moments
Given a random variable  , the  th moment of  is the value ![$ E[X^k]$](http://images.planetmath.org:8080/cache/objects/515/l2h/img4.png "$ E[X^k]$") , if the expectation exists.
Note that the expected value is the first moment of a random variable, and the variance is the second moment minus the first moment squared.
The  th moment of  is usually obtained by using the moment generating function.
Central moments
Given a random variable  , the  th central moment of  is the value
![$ E\big[(X-E[X])^k\big]$](http://images.planetmath.org:8080/cache/objects/515/l2h/img10.png "$ E\big[(X-E[X])^k\big]$") , if the expectation exists. It is denoted by  .
Note that the  and
![$ \mu_2=Var[X]=\sigma^2$](http://images.planetmath.org:8080/cache/objects/515/l2h/img13.png "$ \mu_2=Var[X]=\sigma^2$") . The third central moment divided by the standard deviation cubed is called the skewness  :
The skewness measures how “symmetrical”, or rather, how “skewed”, a distribution is with respect to its mode. A non-zero  means there is some degree of skewness in the distribution. For example,  means that the distribution has a longer positive tail.
The fourth central moment divided by the fourth power of the standard deviation is called the kurtosis  :
The kurtosis measures how “peaked” a distribution is compared to the standard normal distribution. The standard normal distribution has  .  means that the distribution is “flatter” than then standard normal distribution, or platykurtic. On the other hand, a distribution with  can be characterized as being more “peaked” than  , or leptokurtic.
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"moment" is owned by CWoo. [ full author list (2) | owner history (1) ]
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(view preamble)
| Also defines: |
central moment, skewness, kurtosis, platykurtic, leptokurtic |
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Cross-references: standard normal distribution, power, positive, degree, mode, distribution, measures, standard deviation, moment generating function, variance, expectation, random variable
There are 24 references to this entry.
This is version 6 of moment, born on 2001-10-26, modified 2006-09-23.
Object id is 515, canonical name is Moment.
Accessed 18257 times total.
Classification:
| AMS MSC: | 62-00 (Statistics :: General reference works ) | | | 60-00 (Probability theory and stochastic processes :: General reference works ) |
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Pending Errata and Addenda
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