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moment (Definition)

Moments

Given a random variable $ X$, the $ k$th moment of $ X$ is the value $ 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 $ k$th moment of $ X$ is usually obtained by using the moment generating function.

Central moments

Given a random variable $ X$, the $ k$th central moment of $ X$ is the value $ E\big[(X-E[X])^k\big]$, if the expectation exists. It is denoted by $ \mu_k$.

Note that the $ \mu_1=0$ and $ \mu_2=Var[X]=\sigma^2$. The third central moment divided by the standard deviation cubed is called the skewness $ \tau$:

$\displaystyle \tau=\frac{\mu_3}{\sigma^3}$
The skewness measures how “symmetrical”, or rather, how “skewed”, a distribution is with respect to its mode. A non-zero $ \tau$ means there is some degree of skewness in the distribution. For example, $ \tau>0$ 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 $ \kappa$:

$\displaystyle \kappa=\frac{\mu_4}{\sigma^4}$
The kurtosis measures how “peaked” a distribution is compared to the standard normal distribution. The standard normal distribution has $ \kappa=3$. $ \kappa<3$ means that the distribution is “flatter” than then standard normal distribution, or platykurtic. On the other hand, a distribution with $ \kappa>3$ can be characterized as being more “peaked” than $ N(0,1)$, or leptokurtic.



"moment" is owned by CWoo. [ full author list (2) | owner history (1) ]
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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 MSC62-00 (Statistics :: General reference works )
 60-00 (Probability theory and stochastic processes :: General reference works )

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