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# moment

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* $\tau$:

$\tau=\frac{\mu_{3}}{\sigma^{3}}$ |

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* $\kappa$:

$\kappa=\frac{\mu_{4}}{\sigma^{4}}$ |

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*.

## Mathematics Subject Classification

60-00*no label found*62-00

*no label found*81-00

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