# 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.

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.

 Title moment Canonical name Moment Date of creation 2013-03-22 11:53:54 Last modified on 2013-03-22 11:53:54 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 11 Author CWoo (3771) Entry type Definition Classification msc 60-00 Classification msc 62-00 Classification msc 81-00 Defines central moment Defines skewness Defines kurtosis Defines platykurtic Defines leptokurtic